GRAPHS AND HYPERGRAPHS

North-Holland Mathematical Library
Board of Advisory Editors:
M. Artin, H. Bass, J. Eclls, W. Feit, P. J. Frcyd. F. W, Gchring,
H. Halbcrstam, L. V. Httrmander. M. Kac, J. H. B. Kempcrman,
H. A. Lauwcricr, W. A. J. Luxemburg, F. P. Peterson, I. M. Singer
and A. C. Zanncn
VOLUME 6
l<
NORTH-HOLLAND PUBLISHING COMPANY - AMSTERDAM • LONDON
AMERICAN ELSEVIER PUBLISHING COMPANY, INC. - NEW YORK

Graphs and
Hypergraphs
CLAUDE BERGE
University of Paris
Translated by Edward Minieka
r*>r 62890
NORTH-HOI LAND PUBLISHING COMPANY-AMSTERDAM • LONDON
AMERICAN ELSEVIER PUBLISHING COMPANY. INC. - NEW YORK

© NORTH-HOLLAND PUBLISHING COMPANY — 1973
All rights reserved. No part of this publication may be reproduced, stored In a retrieval system
or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording
or otherwise, without the prior permission of the copyright owner
Library of Congress Catalog Card Number: 72-88288
North-Holland ISBN series: 0 7204 2450 X
North-Holland ISBN volume: 0 7204 2453 4
American Elsevler ISBN: 0 444 10399 6
Translation and revhud edition of
GRAPHES ET HYPERGRAPHES
©DUNOD. Paris 1970
First edition 1973
Second, revised edition 1976
.65*8
Publishers:
NORTH-HOLLAND PUBLISHING COMPANY-AMSTERDAM
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PRINTED IN Till-: M-TIIKK1 ANDS

To JEAN-MICHEL

FOREWORD
Graph theory has had an unusual development. Problems involving graphs
first appeared in the mathematical folklore as puzzles (e.g. Konigsberg
bridge problem). Later, graphs appeared in electrical engineering (Kirchhofs
Law), chemistry, psychology and economics before becoming a unified field
of study. Today, graph theory is one of the most flourishing branches of
modern algebra with wide applications to combinatorial problems and to
classical algebraic problems (Group Theory, with Caylcy, Ore, Frucht,
Sabidussi, etc.; Category Theory, with Pultr, Hedrlin, etc.).
Graph theory as a separate entity has had its development shaped largely
by operational researchers preoccupied with practical problems. It was with
these practical problems in mind that we wrote our first book Thiorie des
graphes et ses applications published by Dunod in January 19S8. This text
hoped to unify the various results then scattered through the literature. For
this purpose, we emphasized two major areas.
The first of these areas was the network flow theory of Ford and Fulkerson
which was beginning to transcend analytic techniques. This theory gave new
proofs for more than a dozen graph theory results including some famous
theorems by Konig and by Menger.
The second area was the theory of alternating chains which started with
Pctersen sixty years earlier, but which appeared in optimization problems
only in 1957.
These two areas had many curious similarities; however, the integer linear
programs that they solved did not overlap. Now. more than ever, we believe
that these two areas should form the foundation of graph theory.
The first mathematicians to work in graph theory (in particular the thriving
Hungarian school with D. Konig. P. Erdos, P. Turan, T. Gnllai. G. Haj6s,
etc.) considered mainly undirected graphs, and this could lead students to
believe that there are two theories—one for directed graphs and one for un-
undirected graphs. This book is written with the viewpoint that there is only
one kind of graph (directed) and only one theory for graphs. This is reason-
reasonable because a result for an undirected graph can be interpreted as a result
for a directed graph in which the direction of the arcs does not matter. Con-
vii

viii
versely, a result for a directed graph can be interpreted for an undirected
graph by replacing each edge or the undirected graph with two oppositely
directed arcs with the same endpoints.
Since I9S7, research in graph theory has assumed astonishing proportions.
Results have appeared from all over the world, and some or the conjectures
or our first book have been solved notably by our students in Paris from I9S9
to 1964 (in particular, the late Alain Ghouila-Houri, whose work frequently
appears in this text) and by Soviet mathematicians (in particular, A. A. Zykov,
V. G. Vizing, L. M. Vitavcr, M. K. Goldberg, L. P. Varvak, etc. following the
Russian edition of our first text). Thus, because of this embarrassment of
riches, this book is vastly more extensive, but still cannot treat very specific
applications.'1'
The concept of a matroid due to H. Whitney and developed by W. Tutte
has made possible an axiomatic study of cycles and trees. However, we can-
cannot treat this algebraic aspect of graph theory too extensively without
straying from our purpose. Similarly, new techniques have appeared for
topological graphs, but these would also take us astray. Such strictly topo-
logical problems will be the subject of a future work.13'
On the other hand, this book intends to present a systematic study of the
theory of hypergraphs. A hypergraph is defined to be a family of hyperedges
which arc sets of vertices of cardinality not necessarily 2 (as for graphs).
Given a graph, a hypergraph can be defined by its cliques, or by its spanning
trees or by its cycles. Thus, the theory of hypergraphs can generate simul-
simultaneously several results for graphs.
The formulation of combinatorial problems in terms of hypergraphs often
gives surprisingly simple results that will look very familiar to graph theorists.
At the Balatonfurcd Conference A969). P. Ilrdds and A. Majnal asked us
why we would use hypergraphs for problems that can be also formulated in
terms of graphs. The answer is that by using hypergraphs, one deals with
(') Then ire tcvcrul yruph theory texts that emphualzc opcrationul research problems; in
purticulur: C. Bcrgc und A. Cihoullu-Houri, Programmes. Jaix. ti riseau tie transport. Part II,
Ounod, Paris, 1962(Englishedition. Methuen. London; Wiley, New York. 1963: Gcrmun edition,
Teubncr, Leipzig, 1967: Spuniah edition, Compunia fcdit. Continental, Mexico, 1965): L. R.
Ford and I). R. Fulkenon, flows In Networks Princeton Press, 1962 (French edition. Giulhicr-
Villars, 1967): R. O. Buwcker and T. L. Suuly. Unlit Graphs and Nttworia. McGraw-Hill. 1965;
A. Kaufman, Introduction A la tomblnatortque en rue ties applications, Dunod. 1969 (English
edition to appear): B. Roy. Altttbre motterne el thiorte des graphes. Volume I. Dunod, 1969;
Volume 2.1970: und. Anally. A. A. Zykov. Croph Theory (In Russian). NAU KA Publishing Mowc.
Siberian Brunch. Novosibirsk. 1969.
(')The lopoloiikul nspecta of graph theory will he treined ncpjrilcly in another book to
include such topics an the plannr rcprcscnution or grjphi, genus, thickneiw. crossing number of
non-plunur graphs, proof of the I lea wood conjecture by Rinjiel and Young*. E-'dmonda' methods,
etc.

FOREWORD iX
generalizations or familiar concepts. Thus, hypergraphs can be used to
simplify as well as to generalize.
This English edition contains some results that appeared too late for the
original French edition, especially the Chva'tal existence theorem for hamil-
tonian cycles (Chapter 10) and the Lovdsz proof for the first perfect graph
conjecture (Chapter 20).
An index of all definitions is given at the end of the text so that the .reader
can pass over chapters without much loss of continuity. Theorems are
appended with the name of their first discoverer and the year of discovery.
Sometimes, an old or fundamental result is treated as a corollary, and the
theorem from which it is derived is attributed to a recent author. This is done
purely for didactic purposes and is not intended in any way to diminish the
importance of results that have been generalized. A bibliography arranged
according to chapters is also found at the end of the book.
We first wish to thank Michel Las Vcrgnas and Jean-Claude Fournicr who
have made many notable and original contributions and alterations to this
text, also to Pierre Rosenstichl for the assistance given us during our weekly
meetings. We wish to thank all those who have helped with suggestions, in
particular, J. C. Bcrmond, J. A. Bondy, P. Camion, U. S. R. Murty, 'L.
Lovdsz, J. M. Pla. and W. T. Tuttc.
C. Berge

TABLE OF CONTENTS
PART ONE—GRAPHS
CHAPTER I. BASIC CONCEPTS
1. Graphs 3
2. Basic definitions 5
3. List of symbols ' . . 9
CHAPTER 2. CVCLOMATIC NUMBER
1. Cycles and cocyclcs 12
2. Cycles in a planar graph 17
CHAPTER 3. TREES AND ARBORESCENCES
1. Trees and coirees 24
2. Strongly connected graphs and graphs without circuits .... 28
3. Arborescences 32
4. Injective. functional and semi-functional graphs 36
5. Counting trees 42
CHAPTER 4. PATHS, CENTRES AND DIAMETERS
1. The path problem 55
2. The shortest path problem 59
3. Centres and radii of a quiisi-strong'.y connected graph 61
4. Diameter of a strongly connected jraph 66
5. Counting paths 74
CHAPTER 5. FLOW PROBLEMS
1. The maximum flow problem 76
2. The compatible flow problem 86
3. An algebraic study of flows and tensions 89
4. The maximum tension problem 95
CHAPTER 6. DEGREES AND DEMI-DEGREES
■ ■ Existence of a p-graph with given demi-degrces 102
2. Existence of a p-graph without loops with given demi-degrees . .109
3. Existence of a simple graph with given degrees IIS
xi

Xii TABLB OF CONTENTS
CHAPTER 7. MATCHINGS
1. The maximum matching problem 122
2. The minimum covering problem 129
3. Matchings in bipartite graphs . 131
4. An extension of the Konig theorem 141
5. Counting perfect matchings 142
CHAPTER 8. c-MATCHINGS
1. The maximum c-matching problem .ISO
2. Transfers , ... 153
3. Maximum cardinality of a c-matching ......... 155
CHAPTER 9. CONNECTIVITY
1. ^-Connected graphs 164
2. Articulation vertices and blocks 175
3. fc-Edge-connccted graphs 181
CHAPTER 10. HAMILTONIAN CYCI ES
1. Hamiltonian paths and circuits 186
2. Hamiltonian paths in complete graphs 192
3. Existence theorems for hamiltonian circuits 195
4. Fxistcncc theorems for hamiltoninn cycles 204
5. Hamilton-connected graphs 217
6. Hamiltonian cycles in planar graphs (abstract) 223
CHAPTFR II. COVERING EDGES WITH CHAINS
1. Eulcrian cycles 228
2. Covering edges with disjoint chains 232
3. Counting eulcrian circuits 239
CHAPTER 12. CHROMATIC INDFX
1. Edge colourings .... . ......... 248
2. The Vizing theorem and related results 254
3. Edge colourings of planar graphs (abstract) 267

TABLB OF CONTENTS XIII
CHAPTER 13. STABILITY NUMBER
1. Maximum stable sets 272
2. The Turin theorem and related results 278
3. o-Critical graphs 285
4. Critical vertices and critical edges 297
5. Stability number and vertex coverings by paths 298
CHAPTER 14. KERNELS AND GRUNDY FUNCTIONS
1. Absorption number 303
2. Kernels 307
3. Grundy functions 312
4. Nim games 318
CHAPTER 15. CHROMATIC NUMBER
1. Vertex colourings 325
2. y-Critical graphs ' 338
3. The Haj6s theorem 350
4. Chroma lie polynomials 352
5. Vertex colourings of planar graphs (abstracr) 355
CHAPTER 16. PERFECT GRAPHS
1. Perfect graphs 360
2. Comparability graphs 363
3. Triangulated graphs 368
4. /-Triangulated graphs 369
5. Interval graphs 371
6. Cartesian product and Cartesian sum of simple graphs .... 376
PART TWO—HYPERGRAPHS
CHAPTER 17. HYPERGRAPHS AND TIIE-IK DUALS
1. Hypcrgruphs 389
2. Cycles in a hypcrgraph 391
3. Conformal hypcrgraphs 396
4. Representative graph of a hypcrgraph . . 400
CHAPTER 18. TRANSVERSAIS
1. Matchings ami c-mutchings 414
2. Transversal number 420

Xiv TABU OF CONTENTS
CHAPTER 19. CHROMATIC NUMBER OF A HYPERGRAPH
1. Stability number and chromatic number of a hypergraph .... 428
2. Cliques of a hypergraph 432
3. Good colourings of the edges of a graph 440
4. Generalizations of the chromatic number of a graph 443
CHAPTER 20. BALANCED HYPERGRAPHS AND UNIMODULAR
HYPERGRAPHS
1. Strong chromatic number 448
2. Balanced hypergraphs 450
3. Unimodular hypergraphs 463
4. Stochastic functions 469
CHAPTER 21. MATROIDS
1. Matroid on a set 476
2. The Rado theorem and related results 481
3. Image of a matroid 486
4. Minimum weight basis 493
REFERENCES 498
INDEX OF DEFINITIONS 523

PART ONE
Graphs

CHAPTER 1
Basic Concepts
I. Graphs >;
Intuitively speaking, a graph is a set of points, and a set of arrows, with each
arrow joining one point to another. The points are called the vertices of
the graph, and the arrows are called the arcs of the graph.
The set of vertices of a graph is generally denoted by X, and the set of arcs
of a graph is generally denoted by U. For example, in the graph in Fig. 1.1.
X - { a. *, c, d}, V - { 1. 2, 3,4, 5, 6, 7, 8, 9, 10 } .
Arc 9, which goes from vertex c to vertex d, is said to be of the form (c, d),
for short one may write 9 - (c, d). Arc 2, which goes from vertex a to vertex
b, may similarly be written as 2 - (a, b). Note that arcs 3 and 4 have the
same form as arc 2, but should not be confused with arc 2. In this book, we
shall only consider finite sets X and V. ' ' ~
Note that the position of the vertices in the drawing of the graph is not
important: only the way in which the vertices are joined by arcs is important.
A graph is completely determined by its vertices and by the family of its arcs.
Formally, a graph G is defined to be a pair (X, U), where
A) A' is a set {x,, xa. •■■, xH} of elements called vertices, and
B) U is a family (ux,u2, ...,wB) of elements of the Cartesian product
X x X, called arcs. This family will often be denoted by the set
U ■ { 1, 2,.... m } of its indices. An element (x, y) of X x X can appear
more than once in this family. A graph in which no element of X x A'appears
more than p times is called a p-graph.
The number of vertices in a graph is called the order of the graph.
An arc of G of the form (x, x) is called a loop. For an arc u - (x, y),
vertex x is called its initial endpoint, and vertex y is called its terminal endpoint.
Vertex y is culled a successor of vertex x if there is un arc with x as its initial
endpoint and y as its terminal endpoint. The set of all successors of x is de-
denoted by

ORAPKS
Similarly, vertex y is called a predecessor of vertex x if there is an arc of the
form {y, x). The set of all predecessors of vertex x is denoted by
The set of all neighbours of x is denoted by
Flu. I.I. A 3-graph of order 4
Fig. 1.2. A 1-graph of order 6
Note that r$ is a correspondence from X to X that associates with each
xeXa subsetr£(x) of X. It is possible that r£(x) - 0 (the empty set). If
ro{x) - 0, x is called an isolated vertex. For A <= X, let
ro(A) - U rc(a),
If x 6 ro(/l), x $ A, then x is said to be adjacent to set A,
Forp - 1, a i-graph is called a l-graph. The arcs of a 1-graph are all dis-
distinct elements of the cartesian product X x A". In this case,
Uc X x X,
-m.
A 1-graph G - (A1,1/) is completely defined by X and the correspondence
r - rs. Hence, C can be denoted by (X, T).
In a graph G - (A1, I/), each arc w, - (x,y) determines a continuous line
joining x and y. Such a line, without any specification of its direction, il
called an edge, and is denoted by e, - [x, y).
The family (e,,ea,.... «„) of the edges of G is denoted by its set of indices
£-{1,2 m).

BASIC CONC PTS
If the directions of the arrows in a graph nrc not specified, it is convenient,
for conceptual reasons, to deal with the pair (X, E), rather than the pair
(X. V)- S110*1 a P"'1" (^« ^) '* ca"cd a nwltigraph (or undirected graph).
A multigraph is called a simple graph if:
A) it has no loops,
B) no more than one edge joins any two vertices.
In a simple graph, E denotes a subset of 3(X), the set of all subsets of X
with cardinality 2.
FIr. 1.3. Multigraph FIr. 1.4. Complete graph K» Hr. 1.5. Complete
bipartite graph K3,a
Graphs and multigraphs often appear under other names: sociograms
(psychology), simplexes (topology), electrical networks, organizational
charts, communication networks, family trees, etc. It is often surprising to
learn that these diverse disciplines use the same theorems. The primary pur-
purpose of graph theory was to provide a mathematical tool that can be used in
all these disciplines.
It would be convenient to say that there arc two theories and two kinds of
graphs: directed and undirected. This is not true. All graphs are directed, but
sometimes the direction need not be specified.
Results for directed graphs can be applied to a multigraph G - (X, E) by
replacing G with a directed graph G* that has two oppositely directed arcs cor-
corresponding to each edge in G. Similarly, results for multigraphs can be applied
to a directed graph G — (X, U) after removing the direction from each arc in
G.
2. Basic definitions
A..
Adjacent arcs, adjacent edges. Two arcs (or two edges) are called adjacent if
they have at least one endpoint in common.

6 GRAPHS
Multiplicity. The multiplicity of a pair x, y is defined to be the number of
arcs with initial endpoint x and terminal endpoint y. Denote this number by
«io(x,>), and let
"»<;(*.y) - m'i(y,x),
mG{x, y) - m,;(x, y) + mS(x, y).
If x y, then m^x, y) denotes the number of arcs with both x and y as
endpoints. If x - y, then mc(x,>>) equals twice the number of loops attached
to vertex x. If A and B arc two disjoint subsets of X, let
(A /i) - |{«/u€ (/, m - (xy).
mv{A. B) - <(A «) + m«(ft A).
Arc incident to a vertex. If a vertex x is the initial endpoint of an arc u,
which is not a loop, the arc u is said to be incident out of vertex x. In graph G.
the number of arcs that arc incident out of x plus the number of loops
attached to x is denoted by d$(x) and is called the outer demi-degree of x.
An arc incident into vertex x and the inner demi-degree d£{x) are defined
similarly.
Degree. The degree of vertex x is the number of arcs with x as an endpoint
each loop being counted twice. The degree nf x is denoted by du(x) -
d£(x) + da(x).
If in a graph each vertex h.is the same degree, this graph is said to be
regular.
Arc incident out of a set A<= X. If the initial endpoint of an arc u belongs to
A, and if the terminal endpoint of arc u does not belong to A. then u is said to
be incident out of A, and we write uew'(A). Similarly, we define an arc
incident into A, and the set o)'(A). Finally, the set of arcs incident to A is
denoted by
tu(^) - to*(A) u <o~{A).
Symmetric graph. If mS(x,y) - m5(x,y) for all x,ye X, the graph G is
said to be symmetric. A I-graph G - (X, U) is symmetric if, and only if.
(x,y)eU • (y,x)eU.
Anti-symmetric graph. If for each pair (x, y) e X x X,
'»a(x,y) + mZ(x,y) < I ,

BASIC CONCEPTS 7
then the graph G is said to be anti-symmetric. A 1-graph G - (X, U) is anti-
antisymmetric if, and only if,
(x, y)e U ■» (y,x)iU.
An anti-symmetric 1-graph without its direction is a simple graph.
Complete graph. A graph G is said to be complete if
for all x.^e X, such that * j* y. A 1-graph is complete if, and only if,
(x,y)iU - (y.x)eU.
A simple, complete graph on « vertices is called an n-clique, and is often
denoted by Kn. See Fig. 1.4.
Bipartite graph. A graph is bipartite if its vertices can be partitioned into
two sets Xy and Xa such that no two vertices in the same set are adjacent.
This graph may be written as G ■ (Xlt Xa, U).
Complete bipartite graph. If for all xx e Xx and for all x3e X2, we have
w^x,, xa) > 1, then graph G - (tf,, Xa, U) is said to be a complete bipar-
bipartite graph. A simple, complete bipartite graph with | Xx | - p, and | X2 | - q
is often denoted by A",,,.
Subgraph of 6" generated by A <= X. The subgraph ofG generatedby A is the
graph with H as its vertex set and with all the arcs in G that have both their
endpoints in A.\tG» {X, Pi'\s& 1-graph, then the subgraph generated by
A is the 1-graph GA - {A, rA) where
rA{x) - T(x) n A (xeA).
Partial graph of G generated by V c U. This is the graph (X, V) whose ver-
vertex set is X and whose arc set is V. In other words, it is graph G without the
arcs U - V.
Partial subgraph of G. A partial subgraph of G is the subgraph of a partial
graph of G. For example, if G is the graph of all roads in the United States, the
set of all 4-lane roads is a partial graph of G\ the set of all roads in Illinois is
a subgraph, and the set of all 4-lane roads in Illinois is a partial subgraph.
Chain of length q > 0. A chain is a sequence n - (u,, «a,.... w,) of arcs of
G such that each arc in the sequence has one endpoint in common with its
Predecessor in the sequence and its other endpoint in common with its
successor in the sequence. The number of arcs in the sequence is the length of

8 GRAPHS
chain ft. A chain that does not encounter the same vertex twice is called
elementary. A chain that does not use the same arc twice is called simple.
Path of length q > 0. A path oflength q is a chain yi - (m,, ua,.... u , u,)
in which the terminal endpoint of arc u, is the initial endpoint of arc w, + i Tor
all i < q. For a l-graph, a path is completely determined by the sequence of
vertices xlt x2,... that it encounters. Hence, we often write
Vertex xt is culled the initial endpoint and vertex xktX is called the terminal
endpoint of path /«.
Similarly, for a simple graph, a chain n with endpoints x and y is determined
by the sequence of its vertices, and we may write
] - [x,xltx2 y].
Cycle. A cycle is a chain such that
A) no arc appears twice in the sequence, and
B) the two endpoints of the chain are the same vertex.
Pscudo-cyclc, A pseudo-cycle is a chain ft - (u(, wa,..., uv) whose two end-
points are the same vertex and whose arcs are not necessarily distinct.
Circuit. A circuit is a cycle n - (mi. wa, —, "«) such that for all / < q the
terminal endpoint of u, is the initial endpoint of i/,,,.
Connected graph. A connected graph is a graph that contains a chain
/'[*. y) for each pair x, y of distinct vertices.
Connected component of a graph. Clearly, the relation [x - y, or x * y and
there exists a chain in G connecting x and y] denoted by x ■ y is an equiva-
equivalence relation because
A) x ■ x (reflexivity)
B) x ■ y —■ y ■ x (symmetry)
C) x ■ y,y m z ■» xmz (transitivity)
The classes of this equivalence relation partition X into connected sub-
subgraphs of G called the connected components. For example, the graph in
Fig. 1.2 possesses two connected components.
Articulation set. For a connected graph, a set A of vertices is called an
articulation set (or a cutset) if the subgraph of G generated by X — A is not
connected; the term "cutset" will not be used here, in order to avoid con-
confusion with another kind of "cutset" defined in the theory of transportation
networks (and used in Chapter S).

BASIC CONCEPTS 9
For example, {a, c} and { c} are two articulation sets of the graph in Fig.
I |; vertex r is also called an articulation vertex (or a cut-rertex).
Stable set. A set 5 of vertices is called a stable set if no arc joins two distinct
vertices in 5; for example, {b.d} is a stable set of the graph in Fig. I.I.
Matrix associBtcd with a graph 6'. If G has vertices xitxa xn, let
aj ■ Wo (Xj, Xj) ',
The matrix ((a})) is called the matrix associated with G. For example, the
matrix associated with the graph in Fig. 1.2 is
X i X% X} ^4 A$ ^
0
1
0
0
0
0
1
1
0
0
0
0
0
1
0
1
0
0
0
1
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
1
The matrix Ha})) + ((«'))* is called the adjacency matrix.
3. List
R
N
Z
0
l
U/...
aeA
A\jB
Ac\B
A- B
A s B
A* B
A x B
of symbols
Set of all real numbers.
Set of all non-negative integers.
Set of all integers.
Empty set.
Cardinality of set A. (i.e. number of elements)
Set of all x such that. .
a is an element of set A.
a is not an element of set A.
Union of sets A and B.
Intersection of sets A and B.
A less B (the elements of A that are not in B).
Set A is contained in set B (possibly A - B).
Set A is not contained in set B.
Cartesian product of A and B (the set of all pairs («. 6)where
ae A and be B).
Image of element a in correspondence /*.
Image of set A in correspondence T, or Uo.* /"(«) • (H0) - 0)-

10 GRAPHS
F(A) Transitive closure of correspondence F.
F'X(A) Inverse correspondence of correspondence F, or
(A) Set of all subsets of set A.
. *(/() Set of ull subsets of cardinality k.
. (*,(/() Set of all non-empty subsets of cardinality <&.
A) B) Property A) implies property B).
A) B) Property A) is equivalent to property B).
(n . JL: Binomial coefficient. .
q! q ! (P - /)! »
(I - — p— —. Multinomial coefficient.
p m q (mod. k) Integer p is equal to q modulo k (the remainder from the
division of /> by A is equal to the remainder from the divi-
division of q by k).
log/? Logarithm of p.
[PI Integer part of pi q.
IE]* Smallest integer greater than or equal top/q.
{{a1,)) Matrix whose entry in the /-th row and /-th column is a1,.
Del {(a))) Determinant.
For a graph G,
rc(.v) Set of all neighbours of vertex x.
Faix). Tj(.v) Set of all successors (resp. predecessors) of vertex x.
ctc(x) Degree of vertex x.
(ia(x), do(.x) Outer dcmi-dcgrcc, inner dcmi-dcgrcc of vertex x.
Mt^A, B) Number of edges between sets A and B.
m*,{A, B), ntaiA, B) Number of arcs going from A to B (resp. from B to A).
n[x. y] Portion of the chain n between vertices x and y.
i»(A) Set of all urcs having exactly one endpoint in A.
u>'{A),u>~{A) Set of all arcs with only their initial endpoint in A (resp., with
only their terminal endpoint in A).
For a family / of sets, a member S if is defined to be a minimal set if it
does not contain any other member of ; a member Se if is defined to bo
a minimum set if its cardinality has the minimum value. A maximal set and
a maximum set arc defined similarly.

DASIC CONCEPTS 11
EXERCISES
I Show thnt if G is a simple graph with it vertices and p connected components, the
ninximum possible number of edges In C is
li(n-p)it<-p+ I).
2. Show thnt n simple grnph with n vertices nnd more thnn )(" ~ D (" - 2) edges is
connected.

Chapter 2
Cyclomatic Number
1. Cycles and cocyclcs
In a graph G - (X, V), a cycle is a sequence of arcs
H - (Mi.  w,)
such that
A) each arc uk, where I < k < q. has one endpoint in common with the
preceding urc »<*-!, and the other end point in common with the succeeding
arc !/*,! (i.e., this sequence is a chain),
B) the sequence does not use the same arc twice,
C) the initial vertex and terminal vertex of the chain arc the same.
An elementary cycle is a cycle in which, in addition,
D) no vertex is encountered more than once (except, of course, the initial
vertex which is also the tcrminul vertex).
Given a cycle /«. we denote by /< * the set of all arcs in /< thut are in the direc-
direction that the cycle is traversed, and we denote by //" the set of ull the other
arcs in /<.
If the arcs in G are numbered 1, 2,..., m. then cycle /< is defined by a vector
H - 0*1,^1 /<J.
with
0
+ 1
- 1
if
if
if
tetr.
Henceforth, a cycle and its vector /< will be used interchangeably, and when
we say that cycle /< is the sum of cycles n1 + na. we will mean the vector sum
Property 1. A cycle is the sum of elementary cycles that are pairwise arc~
disjoint.
This is evident because as we traverse. \i an elementary cycle is defined each
time we return to a vertex.
12

CYCLOMATIC NUMBER 13
Property 2. A cycle is elementary if, and only if, it is a minimal cycle. I.e., no
other cycle is properly contained in It.
The proof is obvious.
If A is a non-empty subset of X, io'(A) denotes the set of arcs that have
only their initial endpoint in A and w'{A) denotes the set of arcs that have
only their terminal endpoint in A. Let
A cocycle is defined to be a non-empty set of arcs of the form co(A), parti-
partitioned into two sets oj*(A) and w~{A). Corresponding to each cocycle, there
is a vector
with
10 if itfcoM).
+ 1 if ieu)*{A),
- 1 if ieu)'(A).
A cocycle may be identified by its vector to.
A cocycle is called elementary if it is the set of arcs joining two connected
subgruphs A] and Aa such that
A\ , A2 * 0
AtnA2-0
At u A2 ■ C ,
where C is a connected component of the graph.
A cocircuit is defined to be a cocycle io{A) in which all arcs are directed in
the same direction, i.e. into set A, or out of set A.
Property 3. A cocycle is the sum of elementary cocycles that are pairwi.se arc
disjoint.
Let u) be a cocycle of the form to{A), and let Ax, Aa,..., Ak be the different
connected components of the subgraph generated by A. Then
and the cocycles ti>(/4,). ti>Ma)..... <o{AK) arc pairwise disjoint. It rcmuins to
show that to(/4,) is the sum of elementary disjoint cocycles.
If C is the connected component that contains A,, and if the subgraph
generated by C - A, has connected components C\. C8,.... then

14 ORAPHS
where -co(Ci) is an elementary cocycle since <o<C,) joins the connected sub-
subgraphs Ci and Atu Cau Ca-— Furthermore, -<o(Ci), -co(Ca),... are
pairwisc arc-disjoint. o F n
- Property 4. A cocycle is elementary if. and only if, it is a minimal cocycle (i.e.
no other cocycle is properly contained in the cocycle).
Let (o(A) be a minimal cocycle. Therefore, A is contained in a connected
component C. Let Au A2,.... Ak be the connected components of the sub
graph generated by C - A. If k > 2, the vector -<o(/4i) is a cocycle prop-
properly contained in to(A). But this is impossible, and therefore, k - 1, and to is
an elementary cocycle.
Conversely, let to be an elementary cocycle that joins two connected sub-
subgraphs Ax and Aa. If we remove some, but not all, of the arcs from to, then
we no longer have a cocycle. Therefore to is a minimal cocycle.
Q.E.D.
Arc Colouring Lemma (Minty [I960]). Consider a graph with arcs 1,2,.... m.
Colour arc I black, and arbitrarily colour the remaining arcs red, black or
green. Exactly one of the following conditions holds :
A) there is an elementary cycle containing arc I and only red and black
arcs with the property that all Mack arcs in the cycle have the same
direction,
B) there is an elementary cocycle containing arc I and only green and black
arcs, with the property that all black arcs in the cocycle have the sam
direction.
Successively label the vertices of the graph using the following iterative
procedure:
A) Let arc I - (h, a). Label vertex a,
B) If vertex x is labelled, and vertex y is unlabclled, then label y if
(a) there is a black arc (x, y), or
(b) there is a red arc (*, y) or (y, x).
When the labelling procedure stops, exactly one of the following two cases
occurs:
Case I: Vertex h has been labelled. The vertices used by the procedure to
label b from a constitute tin elementary cycle of red and black arcs with all
black arcs having the same direction. (Thus, there cannot exist a cocycle of
black and green arcs containing arc I with all black arcs in the same direc-
direction.) This cycle is the sum of disjoint elementary cycles, one of which
contains arc 1.
Cash 2: Vertex b has not been labelled. Let A denote the set of all labelleJ

CYC'lOMATir NUMDER 15
vertices. Note th.it to(A) contains only black arcs directed into A or green
arcs. Thus, there exists a cocycle io{A) of green and black arcs containing arc 1
with all black arcs directed into A. (Thus there cannot exist a cycle of redund
black arcs with all black arcs having the same direction). This cocycle is the
sum of disjoint elementary cocyclcs, one of which contains arc I.
Cornllnry. Each arc belongs -ither to an elementary circuit or to an elemen-
elementary codrcuil, hut no arc belongs to holh.
This is shown by applying the lemma with all arcs coloured blnck.
The cycles pa1. ua,.... h" are said to be dependent if there exists a vector
equation of the form
r, u1 + /-, u2 + - + rk fi* - 0 .
where rx, ra rk arc real numbers, not all zero. If the cycles arc not depen-
dependent, they arc said to be independent.
A cycle basis is defined to be a set { u1. hV..., M* } of indcpendenl elemen-
elementary cycles such that any cycle u can be written as
M - r, M1 + fjl>2 + "• + M»x.
where rt, rt rk are real numbers. Clearly, k equals the dimension of the
subspacc of Rm generated by the cycles and therefore docs not depend on
the choice of the basis. This constant k is called the cyclomaifc number of C,
and is denoted by v(G').
A cocycle basis {to', toJ to'} is defined similarly, and its cardinality / is
called the cocycl wtatic number of G and is denoted by A{G).
Exampi e. Consider the graph C in I ig. 2.1; its elementary cycles arc;
M1 -A.6,2) m[abca],
M2 -A.6.3) - [abca],
MJ - B. 3) - [aca].
u4- A.4,5,2) -[aMra],
H* - F. 5.4) - tacdb).
Flu. 2.1 n6 - (|, 4. 5, 3) - [abdea]
These cycles are not independent, since we have, for example:
M1 - n1 + |iJ - 0.
The cycles u2, |i-\ n" form a cycle basis, and therefore, vF") - 3.
A cocycle wM) can be written as a sequence of arcs (±/,, ±i2,...) where

16 GRAPHS
each arc / of the sequence is preceded by a + sign if / e w' (/f) or by a - sign
if ie to'(A), it can ulso be denoted by { A }. For the graph in Fig. 2.1. the
elementary cocycles are:
(,,'.{0} -(+I. + 2.-3).
a* . {ab} -(+ 6, + 2, -3. + 4).
<oJ - {oc} - (- 6, + I, + 5),
w* - {oAc}-(+4, + 5),
w' - {«&/}-(+6, + 2. - 3, - 5),
w6-{««/}-(-6,+ 1,-4).
Obviously, these cocycles are not independent. To form a basis, one could
take, for example, <■>', <o4 and <os; hence /.(G) - 3.
Theorem 1. Let G be a graph with n vertices, m arcs and p connected com-
components. The cardinality of a cycle basis is v(G) -/»-/! + />. The cardinality
of a cocycle basis is ^(C) - n — p.
I. There exist n — p elementary cocycles.
Suppose first thai the graph is connected (p - I). Successively form n - I
independent cocycles ta{Ai), m(Aa) <i>Mn_i) in the following way:
(a) Take an arbitrary vertex a{ and let A, - {a,}. The cocycle <o(/l,)
contains an elementary cocycle. Let [a^ a2] be an edge of this elementary
cocycle such that
a,e A,, a2$ Ax .
(b) Let A2 - A{ u {oa}. The cocycle a>(/f3) contains an elementary
cocycle. Let [x, a3) be an edge in this elementary cocycle such that
xeA2, OiiAt.
(c) Let A3 - A3\J{a3), and repeal the process until n - I elemental
cocycles have been defined.
These cocycles are independent because each contains an arc not contained
in any of the others.
If the graph is not connected (p > I), let C,, Ca C, denote its connec-
connected components. Then, there exist
independent elementary cocycles.
2. There exist m — n + p independent elementary cycles.
Let v(C) - m — n + p, and construct a sequence Co, C, Gn - G of
partial graphs. Graph Go consists of the isolated vertices of G. Each G, is
obtained from its predecessor G,.x by the addition of an arc / of G — G,.i.

CYCLOMATIC NUMDEK
17
Initially- K^o) ■ 0> ant' l^crc arc no cycles. If arc / forms a new cycle (i\
then
nee /> remains unchanged and m is increased by I. If arc / does not form a
new cycle then
since /> decreases by I and m increases by I. Upon termination, v(G) -
m _ n + p cycles m'i. m'» m'« have been defined.
There is no vector of the form
'i M(l + r2 m'1 + - + rk fi1- - 0,
with some rk 0 because cycle n'* contains arc ik which is not contained in
any of the other cycles. Thus the Mi arc r(G) independent cycles.
3. There cannot exist more than v(C) ■/?» — « + /> independent cycles,
and there cannot exist more than /.(G) ■ n — p independent cocycles.
Consider in R the vector space M generated by the cycles and the vector
space il generated by the cocycles. If m is a cycle and if <■> - w(A) is a co-
cycle, their scalar product
equals zero because
<M,w(/l)>-/Mf 1 o(n)\- 1 <M>
\ a* A / it, A
Thus M and 12 are orthogonal subspaccs of R", and their dimensions must
satisfy
dim M + dim Q < wi.
From Parts I and 2,
dim M + dim fi > v(C) + ^(C) - m .
Therefore equality holds throughout, and
dim M - v(G),
dim Ci - /.(G) .
Q.E.D.
*• Cycles in planar graphs
Graph C is said to be planar if it is possible to represent the graph on a
Plane in which the vertices are distinct points, the arcs are simple curves

18
GRAPHS
and no two arcs cross one another. A representation of C that satisfies th'
above requirements on a plane is called a topological planar graph. Tw
topological graphs that can be made to coincide by an elastic deformation o
the plane are considered to be the same.
Example 1. A convex polyhedron in 3-dimensional space defines a simp!
graph: its "corners" (O-dimensional faces) are the vertices and its "sides
A-dimcnsional faces) are the edges of the graph. It has been shown (Steinit
[1922]) that a simple graph Gcan represent a convex polyhedron in Ra if, an
only if, C is a connected planar graph that cannot be disconnected by th
removal of less than three vertices.
Example 2. Problem of three factories and three utilities (Fig. 2.2).
Three factories, a, h, and c rely upon.underground supply lines for thei
water from point </, their gas from point e, and their electricity from point
Is it possible to arrange the three factories and the three utility stations so tha
no supply lines cross one another except at their endpoints? It can be show
that 8 supply lines can always be placed but that the 9th supply line mus
cross at least one other supply line. Thus K3t0 is not planar.
Fig. 2.2
Let C be a topological planar graph. A face of C is defined to be a region o
the plane bounded by arcs such that any two points in a region can
connected by a continuous curve that meets no arcs or vertices. Let Z deno
the set of all faces. The boundary of a face z is the set of all arcs that touc
face z. Faces z and z' are said to be adjacent if their boundaries contain
common arc. (If two faces touch one another only at a vertex, they are no
adjacent.)
The contour of a face z is defined to be an elementary cycle formed with th
edges of the boundary of z that contains in its interior the face z. Note that

NUMItl'R
there is cxiictly one unbounded face and it has no contour. All the other faces
arc bounded and have exactly one contour.
1-xamim.k. A geographic map corresponds to a lopological planar multi-
araph whose edges arc the borders between countries. This graph has no
isthmus, and each of its vertices has degree > 3. A given face inay be adjacent
to another face along several different edges. Note in Fig. 2.3 that faces # and
(I have a common vertex but arc not adjacent.
H(j. 2.3
Theorem 2. In a lopological planar graph C, the contours of the different
bounded faces constitute a cycle basis.
Clearly, the theorem is true if G has only 2 bounded faces. If the theorem is
true for all graphs with/ - 1 hounded faces, we shall show that the theorem
is also true for a lopological planar graph with /'bounded faces.
If not all of the contours arc cilgc-disjoint cycles, then the result is evidently
true. Suppose that when arc / is removed, a graph C with f — I bounded
faces is formed. Hy hypothesis, the contours of this graph C are i fundamen-
fundamental basis of independent cycles. If arc / is returned to the graph, a new finite
face is formed. Its contour is a cycle independent of the cycles of G' because it
contains an arc not present in any cycle of G'. Thus the addition of an arc
cannot increase the cyclomalic number by more than I, and the bounded
faces of C determine a cycle basis.
Q.E.D.
Corollary 1. If a connected topobgical planar graph has n vertices, in arcs
a"dffaces, then
n — m +f m 2 (Eider's Formula).
The number of bounded faces equals the cyclomalic number v(G). Thus

20 GRAPHS
/ - v (G) + I - (m - n + I) + I - m - n + 2.
and the corollary follows.
Corollary 2. A simple planar graph G has a vertex x of degree da(x) < 5
Suppose that G is connected. Otherwise, each connected component will be
considered separately. Since G is a simple graph, each face is bounded by a
least three distinct edges. Consider the bipartite graph Formed by the set
of vertices representing the faecs off/and the set /? of vertices representing the
the edges of G. Place an arc from ae A lobe B each time face a is incident t
edge h. (This graph is called the face-idge incidence graph.) Clearly, the
number of arcs is < 2 m and > 3/. Thus
IT each vertex is the endpoint of at least 6 edges, then n < ~r~. FromLulcr'
o
Formula,
'" 2 m «
2 - n - m + / < y - m + -y - 0.
which yields a contradiction.
Q.E.D.
The Eulcr Formula of Corollary I can be used in many proofs.
Exampi.i: I (liulcr). In 3-dimcnsional space consider a convex polyhcdroi
with it vertices, m edges and/faces. Obviously, one can represent this poly
hedron on the surface of a sphere without having any two edges cross each
other. By a stenographic projection whose centre is the middle of one of th
faces, we can represent the polyhedron on a plane. Therefore this graph i
planar, and we obtain a fundamental relation for convex polyhcdra:
« -/«+/- 2.
Examplf 2. Using Fulcr's Formula, we shall show that the graph for
three factories and three utilities cannot be planar. If the graph is planar, then
/ -2- n + m -2-6 + 9-5.
Each face has at least 4 edges in its contour because if a face s had only 3
edges, then it would be bordered by 3 vertices of which 2 must be in the same
class (factories or utilities), but two vertices of the same class cannot be
adjacent. For the bipartite face-edge incidence graph, the number of arcs i
< 2 m and > 4/. Thus
18 - 2 m > 4/- 20.

CYCLOMATIC NUMBER
21
which is a contradiction.
Example 3. We shall show that the complete graph A'a with 5 vertices
cannot be planar. If this graph is planar, then
/- 2- » + w-2 - 5 + 10-7.
The contour of each face has at least 3 edges. For the bipartite face-edge
incidence graph, the number of arcs is < 2 m and > 3/. Thus
20-2m > 3/"- 21 .
which is a contradiction.
Type I
Type 2
FIR. 2.4
Remark. The graph of factories and utilities and the graph A", allow us to
describe an entire family of non-planar graphs. As shown in Fig. 2.4. unlimi-
unlimited additional vertices may be placed on each edge of these graphs to create
new non-planar graphs of type I und type 2. Conversely it is now possible to
show:
A multlgraph C is planar if and only if. it contains no partial subgraph of
type I or type 2 (Kuralowski [1930]).
Other characterizations of planar graphs have been made by Whitney
[1933], MacLanc [1937]. Ghouila-llouri [1964], W. T. Tuttc [1968], etc.
Algorithms to determine if a graph is plunur have been described by Kcmou-
cron, Malgrangc. Pertuisct [1964], Lcmpcl, Even. Ccderbaum [1967], etc.
Consider a planar inultigraph G that is connected and has no isolated
vertices. Construct a planar multigraph C* thut corresponds to G as follows:
Place a vertex x* of C* inside each face x of G. For each edge e of G, con-
construct an edge e* of G" that joins the vertices corresponding to the faces
separated by edge e. Graph G* is also planar, connected and without isolated
vertices. Sec Fig. 2.5. Graph G* is called the topohgical dual of G. Note that:
A) The topological dual of C* is G. i.e. (C*)* - G.
B) A loop in G corresponds to a pendant edge in G*, and vice versa.

22 GRAPHS
2.5
Theorem 3. I'ach elementary cycle o/G corresponds to an elementary cocycl
of the topofoglcai dual C*, and vice ivrsa.
Let |i - (u,, »a,...) be an elementary cycle of G. and let A* be the set o
vertices of G* that nre inside this cycle. Ilicn
Furthermore. 6\J". is a connected subgraph of C* because we can always g
from one face to another inside cycle |i. Similarly, the complement G*x.-a*
is also a connected subgraph of G*. Thus <n{A*) is an elementary cocycl
The other part of the theorem may be proved similarly.
Q.E.D.
Corollary. For a connected topological planar geaph G.
v(C«) - HG). *(G') - KO •
The corollary is evident from Theorem 3, but it can also be shown using
Euler's Formula, which implies that
v(C») - m -f + I - m - B + m - n) + I - n - 1 - k(G),
k(G*) -/- I - m - n + I - v(G).
EXERCISES
1. Lci w be a set of edges in C. Show ihai to is an elemeniury cocyle if, and only if, w
meeis no elementary cycle or 6'in oniy onearc.and if for each pair of arcs r, e <•>, there
is an eicmeniary cycle n such thai
ftn to •• {e,e'}
(P. Roiensiichl 119701)

CYCL0MAT1C NUMBER 23
2. Consider (he following binary operation, denoted by +, on the set K - {0, I}:.
I +0-0+ I - I,
0 + 0- 1 + 1-0.
A vector |x - (mi. m *»«) e K* is a "topological cycle" if |x - 0 or if bjl represents
the union of disjoint elementary cycles. A "topological cocycle" is defined similarly. Show
that the topological cycles form an abelian group with 0 as the zero clement. Do the same
for topological cocycles.
Show that a vector z e K" is a topological cycle if, and only if, the scalar product
< », w > - 0 for all topological cocycles w. Do the same for topological cocycles.

CHAPTER 3
Trees and Arborescences
1. Trees and cotrees
A tree is defined to be a connected graph without cycles. A tree is a special
kind of l-graph. A forest is defined to be a graph whose connected components
are trees, i.e., a forest is a graph without cycles.
Theorem 1. Let H - (X, V) be a graph of order \ X | - n > 2. The follow
ing properties are equivalent (and each characterizes a tree):
1I) H is connected and has no cycles
B) H has n - I arcs and has no cycles,
C) H is connected and contains exactly n — I arcs,
D) // has no cycles, and if an arc is added to H. exactly one cycle is created,
E) // is connected, and if any arc is removed, the remaining graph is not
connected,
F) Every pair of vertices of H is connected by one and only one chain.
A) =» B) If p denotes the number of connected components, m denotes the
number of arcs and v(H) denotes the cyclomatic number (see Ch. 2),
then (I) implies
p _ | , v{H) mm-n+pmO.
Thus,
m — n — p » n — 1.
B) =» C) Since v(//) - 0, in - n - I, it follows that
p a v{H) - m + n - I .
Thus // is connected.
C) *■ D) Since/) - I, in - n - I, it follows that
v{H) -m-n+pmO.
Thus // contains no cycles. In other words, if an arc is added, the
cyclomatic number becomes equal to I, and there is exactly one
cycle in the new graph.
D) » E) If // were not connected, then two vertices, say a and b, would not
24

TOE S AND ARBORUCENCM 25
be connected, and an arc (a, b) could be added without creating a
cycle, which contradicts D). Thus/? - I, v(//) ■ 0, and therefore
m - n - I.
If an arc is removed, we obtain a graph //' with
rri - ri - 2, v{//') - 0 .
Hence
p' m v(tf') - m' + «' - 2,
and //' is not connected.
E) •> F) For any two vertices a and b there is a chain connecting them (since
// is connected). This chain is unique (otherwise the removal of an
arc which belongs only to the second chain would not disconnect
the graph).
F) *• (I) Clearly, if // had a cycle, at least one pair of vertices would be
joined by two distinct chains, which contradicts F).
Q.E.D.
'I heorcm 2. A vertex is called "pendant "iflt is adjacent to exactly one other
vertex. A tree of order n > 2 has at least two pendant vertices.
Let // be a tree that has only 0 or I pendant vertices. Consider a traveller
who traverses the edges of the graph starting from a pendant vertex (if there is
one). I f he does not permit himself to use the same edge twice, he cannot go to
the same vertex twice (since // has no cycles).
If he arrives at a vertex .v, he can always depart using a new edge (since x is
not a pendant vertex). Thus the trip lasts indefinitely, and this is impossible
since // is finite.
Q.E.D.
Theorem 3. A graph G - {X, V) has a partial graph that is a tree If, and only
If, G Is connected.
If'G is not connected, no partial graph of G is connected. Therefore G can-
cannot have a partial graph that is a tree.
IfG Is connected, look for an arc whose removal does not disconnect the
graph. If no such arc exists, G is a tree by virtue of property E). If such an arc
exists, remove it and look for another such arc, etc.... When no more arcs
can be removed, the remaining graph is a tree whose vertex set is X.
Q.E.D.
The tree obtained from G as above is called a spanning tree, and Theorem 3
yields a simple algorithm to construct a spanning tree of a connected graph.
A spanning tree can also be constructed in the following way:

26
ORAPHS
Consider any arc w0- Find an arc w, that does not form u cycle with t/0.
Then find an arc ua that docs not form a cycle with {w0, u,}, etc.... When the
procedure cannot continue, u spunning tree has been obtained by property D)-
'I hcorem 4. Let Gbea connected graph, I •/ // be a spanning tree ofG, and let
u, be an arc ofG not in tree H. If arc u, is added to If. it creates a cycle \t' by
virtue of property (A). The different cycles ft'form a cycle basis ofG, called the
"basis associated with tree tl".
The cycles n' arc independent since every one of them contains an arc not
contained in any of the others. Moreover, the number of \t' equals:
m(G) - «;(//) - m - (n - I) - m - n + p - v[G).
By Theorem (I. Ch. 2), it follows that the n' form n cycle basis of G.
Q.E.D.
a
II
Fl«. 3.1
Remark. This theorem yields a simple algorithm to construct a cycle basis
of u connected gruph G. If 6' is not connected, euch connected component has
to be treated separately.
Given a connected grnph G — [ X, U). 1 hcorcin I shows that a partial graph
// ■• (X, V) is u spanning tree if it contains no elementary cycles and if upon
the addition of uny arc in U — V, the graph contains an elementary cycle.
Similarly, we shall say that u partial graph (X, W) ofG is u cotrce if it con-

TRMS AND AHHOHISCENCE3 27
tains no clemcntnry cocycles of G and if upon the addition of any arc in
U _ W, it does contain an elementary cocyclc of G.
Theorem 5. LvtG — (X, V) be a connected graph, and let (V, W) be a parti-
partition of V'
V\jW - U, Vn W - 0.
/I necessary and sufficient condition for (X. W) to be a cotree is that (X, V) be a
tree.
I. Sufficiency. If (X. V) is a tree, we shall show that (X. W) is a cotrcc.
W contains no cycles ofG. Suppose that W contains an elementary cocycle
oM) of 6'. Then no chain in the tree (X. V) connects A and X — A. which is a
contradiction.
//" v 6 V. the set of arcs W\j{t>) contains a cocycle of G. Clearly. (X. V -
{ v}) has two connected components A and ii. Therefore. «)(/1) is an element-
elementary cocyclc of G contained in W u { v}.
2. Neccsu'ty. If (.V. W) is a cotrcc. we shall show that (-V. V) is a tree by
using the Arc Colouring Lemma (Chap. 2).
V contains no cycles. Let v€ V. Colour arc r black. Colour the arcs in
V - {v} red. and colour the arcs in W green. Since (-V. W) is a cotrcc. G has
a cocyclc of black anil green arcs that contains arcu. Hi us there cannot be a
cycle of red and black arcs containing arc i>. Since r was selected arbitrarily,
there cannot be any cycle in V.
tj \r e W. the set V u { w} contains a cycle. Colour arc w black. Colour the
arcs in V red. and colour the arcs in W —{>»•} green. Since G has no cocyclc
of black and green arcs containing arc n\ there is a cycle of black and red arcs
containing arc w. This proves that (A\ I ) is a spanning tree of 0".
Q.E.D.
Iheorein 6. Let G" - (Jk\ V) be a connected graph, let F - (.V. H) be a co-
'rce, and let u, be an arc of G not in /•'. // m, is added to /•', it creates exactly
one cocycle w1, and the different cocycles inform a cocycle hasis ofG.
Clearly, if in the graph (X, V - W) an arc u, is removed, exactly two con-
connected components A and B arc formed, and w(/J) - w'.
The cocycles to' arc independent since each of them contains an arc not con-
contained in any of the others. The number of cocycles «' equals the number of
edges in the tree {X. U - W) which equals

28 ORAPHJ
n - 1 - >l(<7).
From Theorem A, Ch. 2), it follows that the to1 form a cocycle basis.
Q.E.D
This theorem yields a simple algorithm to construct a cocycle basis.
2. Strongly connected graphs and graphs without circuits
Consider a connected graph G ■ (A\ (/). A path of length 0 is defined lobe
any sequence [.v] consisting of a single vertex xe X.
For xe X and ye X, let the relation x m y signify that there is a path
nAx>y\ going from x to y and also a path fa[y, x] going from y to x. This
relation is an equivalence relation, i.e. it satisfies the following three properties:
x ■ x for all x,
x ■ y —■ y u x,
xmy, yuz —■ x m z.
The sets of the form
^(*o) - {xlxe X, xm x0}
partition X and are called the strongly connected components ofC.
A graph is said to be strongly connected, if for all x.ye X, there exists
path /j,[jt, ;'] and a path n2[y, x]. In other words, graph G Is strongly connected
If It has only one strongly connect 'd component.
Theorem 7. IfG is a connected graph with at least one arc, the following con-
conditions are equivalent:
A) G is strongly connected.
B) Every arc lies on a circuit,
C) G contains no cocircuits.
A) » B) Let (.v, >■) be an arc of C; since there is a path from >»to x, arc (x, y)
is contuincd in a circuit of G.
B) =• C) If G had a cocircuit that contains arc (x,y), then G cannot have
a circuit containing this arc by the Arc Colouring Lemma with
all arcs coloured blnck. This contradicts B).
C) => (I) Let C be a connected graph without cocircuits. We shall assume
that G is not strongly connected and produce a contradiction.
Since G is not strongly connected, it has more than one strongl)
connected component. Since G is connected, there exist two dis-
distinct strongly connected components that arc joined by an arc

TR11S AND AR0ORESC1NCES 29
(a, b). Arc (a, b) is not contained in any circuit because otherwise
a and b would be in the same strongly connected component. By
the Arc Colouring Lemma, arc {a, b) is contained in some co-
circuit. This contradicts C).
Q.E.D.
Theorem 8. If C is a graph with at least one arc, the following conditions are
equivalent:
A) G is a graph without circuits,
B) Each arc is contained in a cocircuit.
The proof is immediate.
Theorem 9. If C is a strongly connected graph of order n, then G has a cycle
basis o/v«7) circuits.
To prove Theorem 9. it is sufficient to show that v(G) independent circuits
can be found. This is obviously true if G has order < 2. Assume that this is
true for all graphs of order less than n > 2. We shall show that this also is
true for a graph of order n.
Choose from the circuits of length > 1 a circuit \t - (wi, u7 uk) of
minimum length. No arc joins two non-consecutive vertices of this circuit but
there may exist arcs parallel to the arcs of circuit ft. ■
Replace till the vertices of n by n single vertex a'. Replace each arc incident
to n but different from Ui,u3 i/x. by an arc with the same index incident to
a'. The new graph C has order n' m n - k + \ with m' - m - k arcs.
Graph G' is strongly connected and. by virtue of the induction hypothesis,
has a family of v(G') ■ q independent circuits ni, \n'a, <, nj. These circuits in
G' induce in G independent circuits \tlt \ij n . We have
q m v(G') m m' - n' + 1 - (m - k) - (n - k + I) + 1
- v(C) - 1 .
By adding the circuit \t to the family ft,, na pu we obtain a family of
9 + 1 - v(G) independent circuits.
Q.E.D.
Theorem 10. If graph G contains no circuits, then G has a cocyclic basis of
G cocircuits.
We may assume that G is connected. Otherwise, G has several connected
components Ct,Ca C,; the theorem being true for each connected
component, there would be at least
(f (\C,\-l)mn-pm%G)

30 ORAPHS
independent cocircuits, and the theorem would be nlso true Tor G.
It suffices to show that there exists X(G) - n - I independent cocircuttx in a
connected graph G of order n. Clearly, this is true for n < 2. Let n > 2. If this
is true for all graphs with n - 1 vertices, we shall show that it is also true
for n graph G with n vertices.
Since G contains no circuits, there is at least one vertex b such that the
length of the longest path from h equals I, and there exists a vertex a without
successors such thnt (/>, a) e V. Consider the graph G' obtained from G by
deleting the nrcs from /> to a and by replacing a and b by a single vertex a'.
We shall show first that graph G' contains no circuits. If a circuit /i'
were present in 6", it would necessarily pass through a' and hnvc length > 1.
Let d be the vertex that follows a' in this circuit. The cycle n in G induced by
/<' contains arc (a, b) (because G has no circuits). Hence (b, d) - U and there
exists a path of length > I from b to a, which is a contr idiction. Thus G' is
connected, has n - 1 vertices, has no circuits. By the induction hypothesis,
G' has n - 2 independent cocircuits w'(A\), t»'{A'a) w'{A'H.2) and we may
assume
Each of these cocircuits of V induces a cocircuit coM,) in G. Since the
vector w(/f,) has the same coordinates as the vector v)'(A',). the t»M,) arc
linearly independent vectors. The vectors «{</), u>M,), wMa) «>M,,-a)
arc also linearly independent because «)(«) contains the arc (b. a) that is not
contnincd in any of the other cocircuits.
Thus n - I independent cocircuits have been found.
Q.E.D.
Let 0" - (X. V) be a strongly connected graph without loops nnd with
more than one vertex. For each vertex x, there is a path from it and a path
going into it: therefore there exist at least two arcs incident to .v. A vertex that
hns more than two arcs incident to it is called a no<!t\ Otherwise, it is called an
anti-node. A path whose only nodes arc its midpoints is called a branch, A
strongly connected graph without loops that has exactly one node is called a
rosace. A rosace has a very simple structure since each branch leaves from and
returns to the only node.
A graph 0' is said to be minimally connected if it is strongly connected and
the removal of any arc destroys the strongly connected property. Clearly, a
rosace is a minimally connected graph. Furthermore, each minimally con-
connected graph is a 1-graph without loops.
For a graph (i ■ (X, V). the contraction of a set A of vertices is the opera-
operation defined by replacing A by a single vertex a and by replacing each arc

THI'IS AND AHDOKISTKNCn 31
going into A (rcsp. out of A) by an arc with the same index going into a
(rcsp. out of a).
Iimnin I. let G he a minimally connected graph, let A he a set of vertices
that generate.* a strongly connected subgraph of G. Then the contraction of A
yields a minimally connected graph.
1. We shall show first that the contraction of A yields a I-graph. If this
were not the ense, there would exist a vertex x 4 A und two vertices a,a'eA
such that (x, a), {x, a') - U (or, with (a, x), (a\ x) e V but this would not
change the proof). If one of these arcs is removed, the graph remains strongly
connected. Thus, (/ is not minimally connected, which is a contradiction.
2. We shall show now that the contraction of A yields a graph G' that is mini-
minimally connected. Clearly, graph C is strongly connected. If an arc u is re-
removed, the remaining graph is not strongly connected, since the graph
(X, U - { u }) is not strongly connected.
Q.L.D.
I emmn 2. Let G he a minimally connected graph, and let G' be the minimally
connected graph ohtaimd by the contraction of an el •mentary circuit of G.
Then
v(C') + I
Let ft be the elementary circuit to be contracted. It is of length k > I and
has no chords (otherwise G would not be minimally connected). Let n nnd in
respectively denote the number of vertices and arcs in G'. Then
v(C') ■ mi1 - if + I - (m -*)-(«-* + I) + I
- m - n m v(C) - I . Q.E.I),
theorem 11. IfG is a minimally connected graph of order n > 2. then G has
it least two anti-nodes.
Since | X \ > I. C contains at least one circuit, and v(G) > I. If v(C) - I.
the result is true because G is an elementary circuit. We shall nssumc that the
result is true for graphs with n cyclomatic number < k nnd show that it is also
•rue for a graph C with v(G) - k > I.
Case I. G has no circuits of length > 3. Then any two ndjaccnt vertices of
Cnrc necessarily joined in both directions. The simple graph // that ha* the
same vertices as C with two vertices joined by an edge if, and only if. they arc
adjacent in G, is connected (because G is connected) and has no cycles (be-
(because C is minimally connected). Thus, // is a tree of order > 2, and from
Theorem 2, // has two pendant vertices x and y. The vertices x nnd y are
b»th unti-nodes in graph G and this proves the result.

32 GRAPHS
Case 2. G has a circuit /< of length > 3. This circuit has no chord (a chord
is an arc joining two nori-consccutivc vertices of a circuit i if/< had a chord. G
would not be minimally connected). Since v(G') > 2, there exists a vertex of
G that is not contained in /<.
The graph G' obtnincd from G by the contraction of n has order > I, and
from Lemma 2. v(C') - v(G) - I. Because of the induction hypothesis, C
possesses two anti-nodes .v and y. If one of these anti-nodes, suy x, is the
contracted image of//, then /< contains an anti-node z of G (because its length
is > 3). Thus G has at least two anti-nodes, y and :.
If neither of the vertices .v and y is the contracted image of ft, then G has at
least two anti-nodes, x nnd y.
Q.E.D.
Corollary 1. Let G be a minimally connected graph that is not an elementary
circuit. Fhen there exists a branch whose anti-nodes form a non-empty set A such
that the subgraph Gx-a is strongly connected.
Construct From G a graph G* whose vertices are the nodes of G and whose
arcs are the branches of G. Grnph 6* is strongly connected, but it is not mini-
minimally connected (because it has no anti-nodes). Thus, an arc can be removed
from G* without destroying the strong connectivity. This arc of G* is neces-
necessarily a branch of G of length > I, because G is minimally connected.
Q.E.D.
Corollary 2. IfG is a sir mgly connected graph without loops having at least
one node, then there exists a branch whose arcs and anti-nodes can be removed
without destroying the strong connectivity.
The proof is similar to the proof of Corollary 1.
Theorem 12. IJ G - (X, U) is a graph, the graph C obtained front G by
contracting each strongly connected component contains no circuits.
The proof is immediate.
3. Arborcsccnccs
In a graph G - {X, U), a vertex a is called a root if all the vertices of G
can be reached by paths starting from a. A graph does not always have a root.
A graph G is said to be quasi-stronglr connected if for each pair of vertices
A'. ;•. there exists a vertex ;(.v, v) from which there is a path to x and a path
to y. A strongly connected graph is quasi-strongly connected because we can

TRIM AND ARnORCSCVNCIS 33
lei z(x<y) m *• l^e converse is not true. A quasi-strongly connected graph is
connected.
Finally, an arboresconcc is defined as a tree that has a root. For example,
the family tree of the male descendants of King Henry IV is an arborcscence
whose root is King Henry IV.
' y
FIr. 3.2: Arborcscence
Lemma. A necessary and sufficient condition that a graph G - (A', U) have a
root is that G he quasi-strongly connected.
Clearly, if G has a root, then G is quasi-strongly connected. Conversely,
suppose G is quasi-strongly connected, and consider its vertices ,v,, .v xH.
There exists a vertex z2 from which there is a path to .\i and a path to x2.
There exists a vertex z3 from which there is a path to z2 and a pnth to x3, etc.
Also, there exists a vertex zn from which there is a path to zn,, and a path to
xv Clearly, vertex r, is a root of G.
Q.E.D.
Theorem 13. Let H he a graph of order n > I. The following properties are
equivalent (and each characterizes an arhorescence):
A) H is a quasi-strongly connected graph without cycles,
B) // Is a quasi-strongly connected graph and has n - I arcs,
C) // is a tree having a root a,
D) There exixts a vertex a such that each other vertex is connected with it by
one path from a. and only one.
E) // is quasi-xtrongly connected and this property i.<t destroyed if any arc is
removed from //.
F) // ix quasi-strongly connected and has a vertex a such that
d;,(a) - 0,
d7,(x) - 1 (x o).

4 GRAPH*
G) // lias no cycles and contains a vertex a such that
dJi(a) ■ 0
</,7(x) - I (x a).
B) From property (I). // is connected and without cycles. Thus // is a
tree. Therefore. // has n — I arcs.
C) I roni property B), // is connected and has ii - I arcs. Thus // is a
tree. From the lemma. // has a root a.
D) The root a of tree // has the desired property.
E) Suppose that the quasi-strongly connected property is not destroyed
when an arc (x,y) is removed. Then, there exist two elementary
paths
[r, r,,fj ,v] and [r.rf,,rfj y]
that do not use arc (.v. ;•)■ Thus there are two paths in graph G from
z to ;\ and there arc two paths from a to y. This contradicts property
D).
F) From the lemma, graph // has a root a because it is quasi-strongly
connected. Thus
dj,{x) > \ (x t* a).
If a vertex x satisfies </,7(.v) > I. there exist two distinct arcs
ii. r e <■> (.v) and, therefore, there are two distinct paths from a to .
If arc u is removed, the graph still has a root at a and therefore
remains quasi-strongly connected, which contradicts E). Thus
<1ii(x) - 1 (x »* a).
Finally, there cannot exist an are incident into a because the graph
obtained from // by removing this are has a as a root and is quasi-
strongly connected, which contradicts E).
• G) The number of arcs in // equals
d,J(Xj) - n - 1.
Since // is connected and has n - 1 arcs, it is a tree, and contains no
cycles.
(I) Starting from a vertex b a, travel through the graph traversing
the arcs against their direction. No vertex is encountered twice be-
because // has no cycles. If a vertex .v a is encountered, the trip will

TK ES AND AKDOKESCENCES 35
continue because dfi(x) - I. Therefore, the trip can only end at
vertex a. Thus a is a root, and // is quasi-strongly connected.
Q.E.D.
Corollary. A graph G has a partial graph that is an arborescence if, and only
if C is quasi-strongly connected.
If G is not quasi-strongly connected, no partial graph is an arhorescence.
Conversely, if G is quasi-stmngly connected, we can successively delete all
the arcs whose removal docs not destroy the quasi-strongly connected pro-
property. When no such arcs exist, the graph is an arborescence by virtue of
Theorem 13, property E).
Q.E.D.
The following theorem deals with simple graphs and is a constructive ref
formulation of a result of P. Camion [1968].
Theorem 14 (Crcstin [1969]). / ct G - (,V. /:) be a simple connected graph,
and let .v, V. It is possible to direct all the edges of E so that the obtained
graph Go ■ ( V. U) has a spanning tree II such that:
1. // Is an arlmrescenve with root xu
2. The cycles associated with tree II are circuits,
3. The only element try circuits o/ Go are the cycles associated with tree II.
Construct a sequence .v,,.va,... of distinct vertices as follows: Given the
partial sequence .v,, xa .v,, find the vertex x, whose index./ is as large as
possible such that
A) \<J<i.
B) r(l{Xj) * { X,, x2 x,}.
Then take .v,,i e r,:(x,) - { x, .v,}, and direct edge [x,, x,.t] from x, to
•v,, |. Stop when all the vertices arc in the sequence. Let // be the l-graph
formed by the edges directed by [his procedure.
1. l-roin Theorem 13. // is an arborescence with root .v, because // is con-
connected, and
</»(*,)-0.
d;,{x() - 1 for i I .
2. Let [.v,, ,vj. where./ < k, be an edge of £ that is not in //. This edge de-
determines a cycle of the basis associated with //. We shall show that if this
edge is directed from xk to .v,, then the cycle becomes n circuit.

36 GRAPHS
Since xk is adjacent to x,, all vertices x, withy < / < Jc, are in the subar
borcsccncc of// rooted at .^.Therefore, there exists in //a path /<[*,, xk), and
by adding arc (xh, x,), a circuit is obtained.
3. If in arborcscence //. there is a path from .v to y. we write x < y. Clearly
the relation < is transitive. Let
ft - [a,,a2 a, - a,)
be an elementary circuit of Go. Sec Fig. 3.3. Since // is an arborescence.
FIr. 3.3
there exists at least one arc in n, say (a,. aa). that belongs to Go - //. If the
circuit /i is not one of the circuits associated with the tree //. Then it contains
another arc of Co — H. Let (fl,.«in) be the first such arc to occur in fi.
Since /< is an elementary circuit, alti < a2. Moreover, the vertices alt.9.
fli+3 «« cannot be > aa (because /< would pass through a3 twice and
would not be elementary). This contradicts that:
a, - a, > a2 .
Q.E.D.
Remark. The last part of this proof yields easily a result of Chaty A966]:
A strongly connected graph Co has exactly v(C0) elementary circuits, if, and
only if, there exists in Co a spanning tree If such that the elementary cycles
associated with H are circuits.
4. Injcctlvc, functional and scml-functlonal graphs
The concept of an arborescence can be generulized in the following way: A
graph G is said to be infective if </«j(.v) < I for all vertices x. If C is injectivc.

TREES AND ARBORE3C NCES
37
then G is a l-gruph and may be written as G - (A'. F). Then the correspon-
correspondence P is an injectke correspomknee, i.e.
x + y - ax)nf\y)~0.
A graph G is said to be a functional \(dg(x) < I Tor each vertex x. If C is a
functional, then C - (*. T) is a I-graph, and the correspondence f is a func-
function tp defined on X. For example, if A' is a set of states, and if </>(.v) denotes the
unique state that follows x in a deterministic process, then the pair (X.tp) is a
functional graph.
Finally, a l-gruph G ■ (X, O is said to be a semi-functional if
A functional graph is semi-functional. An injective graph G « (X. F) is
semi-functional, because
m- X
Hg. 3.4. Injeciive graph
Fig. 3.5. Functional graph
Property 1. A \-graph G - (X, T) is functional if, and only if, its inverse
I! ~(X,r-l)isinjeciiie.
The proof follows since </S(.v) - </j/(.v).
Property 2. A \-graph G - (X, D is semi-functional if, and only if, its in-
inverse II - (X, /"') is semi-functional.
Suppose that G is semi-functional; let y and / be two vertices of G such
thai /-i( V) n r-\y') 0 and let .v0 6 f»(>') n T^/). We have
0 *

38 OKAPIIS
Therefore. rmi(y) <=■ /"'(/). and equality holds. Thus.
r-x(y) n r-'0'') * 0 - r-'(>■) - r~ V) •
Hence, graph // is semi-functional.
Q.E.D.
If C - (A\ U) is a graph, its adjoint G* is defined to be a I-graph whose
vertices Hj. Ma, ....«,„ represent the arcs of G and which has un arc from
m, to w, ifthc terminal endpoint of the arc in G corresponding to u, is the initial
endpoint of the arc corresponding to u,. A path in G that uses all the arcs
corresponds to a path in G* that uses all the vertices. Thus one can relate
properties on the arcs of G to properties on the vertices of G*.
There arc muny characterizations of adjoints. (Sec llcuchcnne [1964].) The
simplest is given by the following theorem:
Theorem 15. A l-graph II is the adjoint of a graph if. and only if, II is semi
functional.
1. If // - [U, /') is the adjoint of a graph G - (A'. U). then // is semi-
semifunctional, since
r(u) n T(«') ?t 0
implies that in G the terminal endpoints of arcs u and u' coincide, and conse-
consequently,
r(u) - r(u).
2. Let // - (U, F) be a semi-functional graph, and consider the family of
subsets of V of the form
C{u) - { rfl'(r) - r<M)}. or Co> { r/f(i-) - 0
The sets Co, C\. Ca C, of V form a partition of U.
Consider a partition J of U formed by the sets
(/-I. 2 </)
Construct a graph G with vertices xo,xlt xa a-,«i and k arcs from a-, to
a-, denoted by «/,,, u,a utli if
Since r€ and £2 arc two partitions of U, each vertex of // is represented by
exactly one arc of G. Clearly, in G, the terminal endpoint of arc u coincides
with the initial endpoint of arc v if. and only if v e /"(«)•
Therefore. // - G*. Q.E.P.

TREES AND AKB0KE4CENCU 39
fhcorcm 16. An injectwe \-graph G ■ {X, U) is connected if, and only if, G
is quasi-strongly connected.
1. Recall that a graph C is quasi-strongly connected if for each pair x. y e X,
there is i> vertex z whose set of descendants
(i}uf(:)u r\z) u ■■■
contains both x and y. If"G is quasi-strongly connected, it is clearly connected.
2. Let G be injective and connected; then each pair a-, ye X, where x ?* y,
is joined by an elementary chain
/'[•*.>"] - [x,at,a2 ak,y).
First, suppose that (.v. f/je U: then
(x,a{)eU m. (ak,y)eU
because, otherwise, there exists an a, with ds(a,) > I. Thus the vertex
z(x, y) - a- has both a- and y for descendants.
Now suppose that (>". «*) e U\ then
(y,ak)eU ~ (allX)eU.
Thus vertex z(x, y) - y has both x and y for descendants.
Finally, suppose (a,, .r) £ U and (ok, >■) (./; then there exists at least one
vertex a, in the chain fi[x, y] such thnt
(o,.fl,.,)e t/, (fl|,fl(+iNt/.
Since C is injective. fi[at,x] Jiid ;i[fl,,;•] arc paths, and z(x,y) - a, has both
x and y for descendants.
This shows that G is quasi-strongly connected.
Q.E.D.
Corollary. A functional \-graph G - (X, T) is connected if, and only if, the
inverse I -graph (A'. /'"') is quasi-strongiy connected.
The proof is obvious.
Ihcorem 17. A necessary and sufficient condition that the edges of a simple
graph (i — (X. li) can he directed to form an injective (resp. functional) graph is
that each connected component ofG contains at most one cycle.
I. Necessity. Let // be an injective graph: each cycle is a circuit (otherwise,
tlicre would be two arcs leaving the same vertex). No two distinct circuits can
have a common vertex because then two arcs would enter the same vertex.
On the other hand no two circuits in the same connected component can be
without a common vertex, because each arc incident to the circuit is directed
°"t of the circuit (and there is no vertex r that has descendants in both
circuits).

40 ORAPHS
Thus there is at most one cycle in each connected component.
2. Sufficiency. Let G - (X, E)bca simple graph with at most one cycle in
each connected component. Direct its edges in the following way: I fa connec-
connected component or G contains no cycles, it is a tree, and its arcs can be directed
to form an arborescence. If a connected component of G has only one cycle,
direct first the edges of this cycle so that it becomes a circuit. Then, by con-
contracting into a single vertex x0 all the vertices in the cycle, the connected com
ponent becomes a tree: direct the edges of this tree to form an arborescence
with root x0.
Clearly, the directed edges induce in G an injective graph.
Q.ED.
Theorem 18. Let <p be a mapping from a subset of X Into X. Each connected
component Cx of the functional graph G - (A\ <p) is the union of two connecte
components Dx and D^<xl of the functional graph H - (X, qP)^
Moreover, if Dx ■ D^^, then Dx is a connected component of II with a
circuit of odd length.
Let Cx denote the connected component of G that contains x.
1. If ye C,, there exist from Theorem 16 integers/? and q with (p*q(y) -
<pa'(x) or <p*(y) - <plr*l{x). Let s(x) - <p\x). We may write
either g<(y) - g>(x) or g"(y) - g'[v(x)].
Hence
yeDx\jD«x).
Conversely, y e Dx u Z>Wx) implies that <p"(x) - <p"(y). and therefore y e Cx.
2. If Dx - £„(»), there exist integers p and q with
or, equivalcntly,
This proves the existence in G of two paths n[x, z] and v[x, z], one of which
is odd and the other even. Hence, there exists a cycle of odd length. Since G is
functional, this cycle is a circuit, and its vertices are the vertices of a circuit in
graph // - (JT. q>%
Q.E.D.
Application (Rufus Isaacs). Find a real valued function <p(x) on R such
that <!>[<p(x)] - ax + b, where a. b e R.

TREES AND ARBORiSCENCES
41
If a > 0, Mcngcr found:
1 +
This function <p is the required solution, since
i b rl b
- ax + b.
If a < 0. the problem is more difficult. For example, take a - - I and
6-0. From the preceding theorem, we know that a connected component
of the graph C - (X, </>) is the union of two distinct connected components of
//, for example. Dx and Dx,x if at [2 A:, 2 k + I] or Dx and Z)x_, if
* [2 A + I. 2 k + 2]. This immediately gives the graph of the function tp
(shown by the dotted lines in Fig. 3.6).
-x-\f -
Fi«. 3.6. Two connected component* of Ihe graph {X. 9>»)
\
t
• A
PI
■♦.V
-I
/-■*
/ -3
2
3.7. Funciion <p such that p*(x) - -x

42 GRAPHS
5. Counting trees
Before presenting results about the number ofdiiTerent trees in a graph, we
shall state some properties of multinomials coeflicients:
By convention, these coefficients will be cquul to 0 if we do not have
nit na,.... np > 0 and «, + n2 + — + np - n.
Proposition 1. Let X be a set ofn distinct objects. Let n{, «a, ...,nf be non-
negative integers such that nx + n2 + ••■ + np •■ n. The number of ways to
place the n objects intop boxes X{, Xa,..., Xp, containing «!, «a np objects
respectively, is
11
The set Xt can be chosen in ( ") different ways. Suppose the set X, i
chosen, then the set Xa can be chosen in j" ~ "'I different ways, etc. Hence
the required number is
(n\/«- nt\ In — «| - n2\ __ (itp\
nj \ n2 I \ "a / \"p/ "
n ! {n — M|)! (n — n\ — n2)! "p
Mi I (w ■■ fl|)! m2 ! (n ■- M| ■■ m2) ! Mj I (n — W| ■■ m2 — nj •) nft
n!
Proposition 2 (multinomial formula). Given p teal numbers
fli,fl2 areK,
•fp have
(«. + «2 + - + flp)" - I („ " i( W1 (a,) - («,
ih.«JhTTii,»o\«Ii «H •■•■ "p/
Consider real variables a), where I < / < «. I <y < />. and form the
product
(rt{ + a\ + •» + or) (a] + a\ + - + a\)... (a? + a\ + - + flj).

TRE S AND AKBOKESCENCES 43
Given the integers «i,na, —,np whose sum is n, consider in the above
polynomial a monomial of the form
(a, a, ... a,') (a2 a2 ... a2')... (a, a, ... a, ')•
This monomial corresponds uniquely to an arrangement of the set
/V - { 1, 2 n} into the boxes NXt Wa Np, where
I AM -»i.l^al-«a W-rt,.
By proposition 1, the total number of such monomials is therefore
f « ) "i .
\"i, ttit ••■! "p/ M| ! n21... np l
If we put
a,1 . flB - «• . a" - a,
for all /, we obtain the desired formula.
Q.E.D.
Proposition 3.
( " )- i ( " V
\'l|i »2< •••• "j»/ l/n,»l \"l« n2> ■•■' "l-1 • "l ~ ' i "l+l' •■•• "p'
Clearly,
(o, + a, + - + a/ - (a, + a2 + - + a,)(fl, + a2 + ■•• + a,)" ,
and the general term of this polynomial is
L n" n)a'il «?-«?-
\iii n2, ..., npl
</n,#0 \"l» •■•» "j — I, .... Mp/
Q.E.D.
We are now ready to consider the problem ofeounting the number of ways
to choose a set E <=■ ?{X) such that the simple graph (X, E) is a tree.
Theorem 19. Let T(n; </,, cla,.... </„) dvmne the number of distinct trees H
with vertices ,V|. xa .vn and with degrees d,,(Xi) - dt. rfH(.va) - d*
</,,(.vn) - dH. Then
(n — 2 \
1. Clearly, the sum of the degrees is twice the number of edges. From
Theorem 1. the sum of ihc degrees for a tree is 2 (« - 1). Thus. 7' 0 only if

44 ORAPHS
£ W, - 1) - 2(n - 1) - n - n - 2.
Without loss of generality, we may suppose that dt > da > •- > dn\ since
the above equality implies that </„ - 1, vertex xn is pendant in the tree.
2. We shall show that
T{n\dud2 <*„)- T T(n-\;dltd2 d, - 1 <*,,_,).
Let C, be the set of the trees // with vertices xltxa, ...,xH and degrees
rf/fUft) ■ ^k> suc^ l^at the pendant vertex xn is joined to xt. If </, > 2, then
Hr,|-7Xn-l ;</,,</, </,-l </„_,).
Since the set of nil trees is the union of the sets , for d, > 2, the above
equality follows.
3. The theorem is true for // - 2. Assume that n > 3 and that the theorem
is true for n — 1. Then
nn;dltd2 </„)- £ T(n-\;dltd2 d, - 1 </„.,)-
-f " V
W, - \,d2- 1 d.- 1/
Q.E.D.
Corollary 1 (Caylcy [1897]). The number of different trees with vertices
i,*a xn isn*-2.
Using Proposition 2, the number of trees cqunls
Corollary 2 (Clarke [1958]). The number of different trees H with vertic
i, jfa,.... xn and with dH(xt) m k is
n - I)""* .

TRIES AND ARBORESCINCU 45
The desired number equals
v ( n~2 U
<£...*. \k- Ud2- \,d3- \ dn - 1 /
(n - 2) 1 v / n-k-l \
" (k - !)!(« - k - 1I MJx*i \d2 - l,d, - 1 dH -1/
(by setting all variables equal to I in the multinomial formula).
Corollary 3 (Moon [1967]). Let G ■ (X, E) be a simple complete graph of
order n. Let (*,, Xt X,) be a partition of X, and let
tl m— / Y P \ V — IV V \ o _/v |r J
"I "■ I" 11 **U*  m \*2* ^l)* •••« "p m \Apt\£pFt
bepairwise disjoint trees of orders \ X, \ - nt. The number of spanning trees of
G that have Hx, //a,.... Hpas subgraphs is
nHltH2 H,) - n, n2 ... n, n'.
If each set X, were contracted to a unique vertex a,, then, the number of
trees /7with
dr{(a,)-d, (/ - 1, 2 p)
is
<*, - l,d2- 1 dp- 1/"
To each tree R correspond exactly («i)di (n2Y*... (np)d* different spanning
trees //of graph G. Hence
(rf 1 /l rf
\fl| - 1, a2 - 1, ..., a, -
■ «i n2 ... np(nt + n2 + ••• + n,)'.
Q.E.D.
Corollary 4 (Caylcy [1889)). 77ie /imw^ of forests with vertices xx,x
d
) f f x
and with p connected components such that xx,x xf belong to p
different trees Is

46 GRAPHS
Let C be the set of trees // on the vertices x0, x,, xa,..., xR such that d,,(x0)
p. From Corollary 2,
If/»c { 1,2 n)and\P\ - p, let W, denote the set of trees inr such thai,
for all leP, the vertex x, is joined to x0. Then
Hence
Therefore,
1 (p-!)!(»- P)! «!
Q.E.I)
Let A' - {X|, x xn} be a set of n vertices, and let E c a(x) be a set
of q edges that join pairs of vertices in X. We now propose to calculate the
number T(X, if) of dilTerent trees on X that do not contain any edge of
Let (X. f) be a grnph with n vertices, q edges and p connected components
with, respectively, «i, na nr vertices. Let
I 0 if graph (X, F) contains a cycle,
I /11 /i2 •■■ "p otherwise.
Theorem 20 (Tcmpcrlcy [1964)). l'he number of different trees on set X that
do not contain any edge in F. is
Iff 6 E, let At denote the set of all trees that contain edge e. Let F c E. If
{X, F) has no cycles and hasp connected components. Corollary 3 to Theorem
19 states that the number of dilTerent trees that contain all the edges of F\

TREHS AND ARHORHSCHNCCS 47
If (X, F) contains a cycle, the above formula is still vnlid, since both sides of
the equality arc 0. From Sylvester's Formula, we have
r * a
Q.E.D.
Corollary 1 (Wcinberg [1958]). If Els a set ofq pairwise disjoint edges, then
In this case, if F £ £, then
and
Q.E.D.
Corollary 2 (O'Ncil [1963]). If E is a set ofq edges, all with a common end-
point x{, then
The proof follows, since we have
\ n! \ n n!
Q.E.D.
Corollary 3 /*•/ S c X with \ S \ - s. IJ E is the set of edges that join all
w posxihle pairs of vertices in S [i.e. [S, E) is a complete graph), then
T(X,E)-n"-2(l -~j' \

48 OKAPHS
Let „ denote the family of subsets Fc£ such that E, F) is acyclic and
has p connected components. Then
For P c 5, | P\ - p, and Fe fp, consider the triples E, F, P) such that the
graph [S. F) has a vertex of P in each connected component. From Theorem
19. Corollary 4.
| {E. F.
Therefore
v
-|{(S.F,P)/Pc S.|/»|-p
-,?.»"'" s"'"'
Consequently,
Q.E.D.
Corollary 4 (Scoin [1962]). If graph (X, E) is the union of two disjoint com-
complete graphs E. V) and(T, W) with \ S| - J and \ T\ - f. then
T(X,E)-s'-1 i»-«.
From Theorem 19. it follows that
T(X,VuW) m T(X,V) m T(X, W)
Therefore, by Corollary 3,
<■•<■>■
Q.E.D.

TRIM) AND AKROKFSCENCCS 49
Corollary 5 (Moon [1%7]). If E is a set of m - 1 edges that forms an ele-
elementary chain on a set Y of m vertices, then
m - p !\ n
If fc E determines a graph (Y, F) with p connected components, then
|F|-| Y\-p -m-p.
i. '"a '"p are respectively the numbers of vertices of these connected
components, then w, + m2 + ■•• + mp ■ m. For | F\ - m — p, there arc
as many graphs (Y, F) as there arc ways to choose positive integers mi,, mit
.... mp that sum to m. Thus
-II,,)
p-l \ " /
The last summation equals the coefficient of xm in the expansion of
(* + 2 x2 + 3 x3 + •■■)' - *p(l - x)'.
From the binomial formula, this coefficient equals
„,.. - 2 />(- 2 p - 1)(- 2 p - 2)... (- 2 p - (m - p - 1))_
*~1J (m-p)l
+ p - 2)... B p + 1) 2 p _ (m + /> - l\
(w - p)! "\ m - p } '
(w + p - 1) (m
(w
Q.E.D.
We now turn to the problem of counting the partial subgraphs of a given
h G that arc urborcsccnccs.
Let A - ((<?J)) be the matrix associated with graph G, where
denotes the number of arcs in G from x, to x,. Let D ■ ((</,')) be the diagonal
matrix defined by
d,l-0 if / J
M-«g(X-{*i}.*i) if '->•

50 ORAPHS
The matrix D — A ■ ((</{ - o{)) can be written as
Ifl'. -«i ...
D - A m
I-i
1*2
-a?
- fl,
- fl.
- a.
- a:
*»
The dctcrminunt of this matrix equals 0 because the sum in each row is 0. The
minor obtained by removing the first row and first column of this matrix is
denoted by
Al -
Lemma. Let G - (X. U) be a grapli with m - n — 1 arcs ami no loops.
Then C Is an arborescence with root x% if. ami only if, A% - I. Otherwise
Ax -0.
1. If C is an arborescence with root *,. then ii\ is equal to dS(x,) ■ 1 for
/ ■ 2, 3,.... n. Index the vertices so thut the indices increase along any path
(this is possible since C is an arborcsccncc). Then
I
-al
-a\
-al
-al
-al ..
-al ..
-fll ..
■ -al
• -al
• £/■
1
0
0
-«2
i
0
-«2
i
-al
0
0
1
- 1
2. We shall now show hy induction on n thut if C is a gruph with n ver-
vertices, m - n — 1 arcs and with A, 0. then G is an arborescence with root xt
Each vertex xk, where k 1. is the terminal endpoint of at least one arc of
G (because, otherwise, the A-th column vector of Ay is the zero vector and
A i - 0). Since wi - n — 1, the inner demi-degrees (we Chapter 1) satisfy the
equalities:

THUS AND AKIIOKISCINffS
SI
(i)
tl,~;(XK) - I {km 2.3
1 Icikc. by Property (ft) of Theorem 13. ii is sufficient to show that G is connec-
connected in order to show that 0' is on arborcscencc. Suppose that G is not con-
connected. 1'hen I, can be decomposed into two square matrices li' and li' in the
following way:
A,
T{
0
I'rom equation (i). we sec that the vertices .v, with s S anil the vertex ,v,
generate a subgraph G' with m' ■ ri — I. The vertices .vr with le Tand the
vertex .V| generate a subgraph G" with in" - n" — 1. Since Oet (fl") 0 and
Del (/?") 0. we know by the induction hypothesis that G' and G" urc
arhorcsccnccs with root .v,. This contradicts the assumption that G is not
connected.
O.F.U.
theorem 21 (Tutte [1948]). Let G - (A\ U) he a graph, amllet *, X. The
number of partial subgraphs of G that are arhorescences with root vt equals a^
i*i
1-1
I
, x,).
Without loss of generality, we may assuinc that G has no loops. Note that
i ■ ^i(a2, a:< .... an)isa linear function of the last n — I column vectors of
c matrix ((«})). i.e.,
^(« + ■«O ■ l(
, a.)
, an)
denote by ck - @,0 1.0 0) the n-vector with its *-th coordinate
Ctl«al to I and with all other n - 1 coordinates equal to 0. Then

52
GRAPHS
1.e»> cj.
From the lemma, Ai(cka, ckj,..., ckll) equals 0 or 1. For each term in the sum
the partial grapli defined by
has no loops (because G has no loops) and is Formed from the n — 1 a ret
Finully, From the lemma, we know that -d,(ckB. ckl,.,., c^J - I if, and onl
if. the graph is an arborcscence with root .\i. This completes the proof,
Q.E.D.
Corollary. IfG — (X, E) is a simple graph, the number of spanning trees Ai
6* is equal to the minor (which is independent of the coefficients of the pnncipa'
diagonal) of the square matrix ((/>})) of order n. where
-d0
- -
-0
1
if
if
if
/
i
i
"J
+ J
j
and
and
[X|,
x,L
E
E
Let C* be the grapli obtained from G by replacing each edge by two oppo
sitcly directed arcs. A spanning tree of C corresponds uniquely to an arbore
cence of G* rooted nt *| (say). Therefore, by Theorem 21. the number o
spanning trees in G equals
b\
bl
Q.E.D.
ExAMPt.t;. Consider the graph G in Fig. 3.8. As shown in Fig. 3.9. th
number of spanning trees in G equals
3 -I -I
1 3 -I
1 -1 3
- 16.

TRIfS AND AHDORESCCNCES
S3
Mr. 3.8
«* 3.9
EXERCISES
•■Let m and n be two integers.
(I) State a necessary and sufficient condition for the existence of a strongly connected
?!P With " vcniccj "nd
?r!e " cniccj "nd '" *<*&*■
low k lhal" for a" stron«'y connected I-graphs with n vertices and m edges, the
wer bound on the number of edges whose removal can destroy the strong connectivity
»[:]■
l- Show that if C - (X. E) is a simple graph such lhat each edge Is contained in some
I mentttry eyc'c. then the nntl-symmctrie graph C conuructed from C ns in Theorem 14
wrongly connccied; show thnt C has only m - n + I - iKO elementary cycles, nnd

54 GRAPHS
ihai a graph without circuits can be consirucicd from G" by reversing the direction of i
subset of its arcs with cardinality
min { m - » + I, n — I ).
(Chaty [1968)
3. Show thai the number or trees on n vertices ihai have exactly k pendnnt vcrti ,
" ' X"z\, where 5; denotes the Slirling number of ihe second kind.
(Rcnyi [19591
4. Consider a &ci X ■ {x^.x xn) and ihe group SH of pcrnuitnnons on X. A •
7" of transpositions \x,..«,] defines a simple grnph {X, if"). Show that:
A) A set To( n — I transpositions gcncrnies the symmetric group 5. if find only if ih
graph [X. T) Is a tree.
B) A5cn6s) If /is a circular permutation of degree n. then the number of ways '
writing/as a product of n — I transpositions equals ft"~s. (Use Corollary 2. Theorem 19
Tor n detailed proof, see C. Herge. Principles of Combinatorics, Academic Press, Ne
York. 1971, p. 143.)
5. A circulation tree of n connected graph C - (X, U) is dcltncd as a spanning tree sucl
that each associated cycle is a circuto of C. Show thni a strongly connccied graph with
vertices and m arcs has a circulation tree if. and only if, the toial number of clcmcntar
circuits is m - n + I. (Chaly A971]

CHAPTER 4
Paths, Centres and Diameters
1. The path problem
The path problem is the following: Find (as quickly as possible) a path from
agiren cert ex a to a given vertex b In a I -graph G ■ (A', U). The chain problem
is similarly defined for a simple graph G - (X. L). Note that the chain prob-
problem becomes a path problem in the I-graph G* - (A'. U) obtained from
G m {X, £) by replacing each edge in £ by two oppositely directed arcs.
iixAMPi.1:. Problem of the hunter, wolf and Brussels sprout. A hunter, a
wolf jikH a Brussels sprout arrive simultaneously at a river bank. The ferry
boat is too small to take more than one passenger (in addition to the boatman)
at the same time. For obvious reasons, the boatman cannot leave the hunter
and the wolf alone together, nor can he leave the wolf and the Brussels sprout
alone together. How should he arrange their passage across the river?
Phis well-known problem can be solved mentally by considering only a
small number of states. Nonetheless, it is a typical example of the path
problem. A graph of the various states can be constructed, and a path must
be found from state a (the hunter //. wolf IF. Brussels sprout S and boatman
flare all on the right bank) to state b (all arc on the left bank). One solution to
the problem is shown in Fig. 4.1.
0
''°r more complicated cases, several systematic algorithms have been pro-
P°scd. If the graph is already known, it is always possible to find all the clc-
mcntary paths starling from vertex « by constructing all the dilTercnt arbo-
55

56 GRAPHS
rcscences rooted at a. Such an arborescence for the above example is showr
in Fig. 4.2.
HWB
-•H
S WSB
Fig. 4.2
By inspection we sec that there are two paths from a to b.
A formal statement of an algorithm for finding all elementary paths from
was given by B. Roy [I960], and others.
Of more interest arc local algorithms applicable when the entire graph is not
known. Ideally, a local algorithm will not trace through the entire graph.
Let G - (X, U) be a graph, and let ae X. If m,, u3 um arc the arcs of <3,
define an alphabet whose letters are + m,, + «a,..., + wn, - w,, — ua, .
- um (positive and negative letters). A word is a sequence of letters, written
in the form
l>! + V2 + — + f| + fj+l + ■" + "k •
If V|*i - - i>i, the word
is called a reduction of the preceding word.
A word nk - i>, + i'a +•••+ vk is called the trajectory of graph G when:
A) r, is an urc with a as its initiul endpoint, and tracing through the
arcs corresponding to the consecutive letters of a word (in the direction of the
arc if the letter is positive, and in the opposite direction if the letter is negative)
defines a chain ^k,
B) If u V. then nk contains the letters + u und - w at most one time
each.
C) If the letter - u is in the word, then the letter + u precedes - u in the
sequence.
D) After word n" has been reduced as much as possible, the reduced word
contains no more negative letters (and, therefore, the reduced word i'tk define!
an elementary path starting at a).
The Trdmaux Algorithm given below is a rapid, local algorithm to con*
struct a trajectory terminating at vertex b.

PATHS, CENTRES AND DIAMETERS 57
algorithm
We shall construct successively a sequence of the trajectories /*', /t3
Each time an arc is used or a vertex is encountered, it is labelled. The algo-
algorithm consists of three rules:
Rulu I. Label vertex a. If u - (a, x) is an arc with initial end point a, put
.,» a + u, (In other words, we advance along arc u and label it.)
Rule 2. Let nl - i'i + i>a +■■■+ «'i be a trajectory terminating at jc with
D, > 0. If vertex x has not been previously encountered, and if x is the initial
endpoint of an arc w, »t - (*, y), let /«'*' - n* + m, + i and label arc w,.t and
vertex;'. Otherwise, let /jim - /«' - 0,, and label arc i1, again.
Rule 3. Let rf - i>i + —+ t, be a trajectory terminuting at .v with v, < 0.
If x is the initial endpoint of an arc w, M - (jr. y) $ /«', let fiu i - nl + u,,,.
Otherwise, let /il + l - /«' - u, where w, is the last letter of the reduced word
Trfonaux's theorem. When the above algorithm terminal 'J. the terminal tra-
trajectory y" has the property that for eaeh arc u in a path starting from a, ^*
contains the letters + u and — u exactly once.
1. When the procedure ends, the trajectory has returned to a, and all the
arcs incident out of a have been labelled (by Rule 3).
2. We shall show that all urcs u labelled by the procedure have been
traversed in reverse. In fact, if,each time the procedure reaches a labelled ver-
vertex via an unlubclled arc. we detach the arc from its tcrminul endpoint
before traversing it in reverse (Rule 2). the graph of labelled arcs would
become an arboresccncc rooted at a (by Theorem 13, Ch. 3. Property F)).
Hence, if this arborcsccnce is explored through the same trajectories, all its
arcs will have been traversed in reverse.
3. We shall show that if there exists an elementary path from a. say
V - («,,!!, Uf) - [fl.fl,.fl, ar).
then the urc up has been labelled by the procedure.
If not, then vertex ap.x has not been labelled (because, otherwise the arc
preceding vertex ap.x in the trajectory has not been traversed in reverse by
cither Rule 2 or Rule 3. which contradicts Part 2 of this proof). Therefore,
''■"cup., has not been labelled. For the same reason,arc wp.a has not been
labelled, and arc w, hus not been labelled. But this contradicts Part I of this
Proof.
Q.E.D.

58 GRAPHS
Remark /. With Tremaux's Algorithm, a path from a to b will be found b
purely local methods, if such a path exists. In fact, the procedure neve
labels the same arc more than twice—once in each direction. This givesq
bound on the number of steps in the algorithm.
Remark 2. If a chain between two vertices in graph G - (AT. LI) is sought,
the algorithm can be applied to the symmetric graph G* constructed from
by replacing each edge in G by two oppositely directed arcs. To avoid the
possibility of using an edge of G four times, we can. after labelling an arc in
for the first time, remove the oppositely directed arc from the graph. This can
also simplify the procedure.
Remark 3. For a graph G - (X, U), it is easy to see that the Trcmaux
Algorithm described above can be used to construct a maximal arborcscence
rooted at a given vertex. This arborescence is defined by the set of all labelled
arcs which have been traversed in reverse by using only Rule 2.
Algorithm (P. Roscnstichl. J. C. Bcrniond)
P. Roscnstiehl [1966] noted that the Tnimaux Algorithm is a special case
of a class of algorithms described by Tarry [1895], He also devised another
labelling procedure, called the algorithm for new arcs, which is described
below.
We shall successively construct the trajectories n\ na,... by the following
rules:
Rule I. If u — («. x) is an arc whose initial endpoint is a, let nl — + u.
In other words, advance along arc u and label it.
Rui r 2. I ct /i1 ■ r, + •■■ + r, Ik a trajectory with x as terminal vertex. If x
is the initial endpoint of an unlabcllcd arc h, ,, - (.v. ,»•). let n' *l - ft' + u, ■
Label the arc ulfl. If there docs not exist an unLibcllcd arc with initial end-
point x, and if the reduced word £, is not null, let p1*1 ■ //' — u. where u
is the last letter of the reduced word.
Theorem. When the above algorithm t •rmfnates. the terminal trajectory /r
has tin property that for each arc u in a path from a. ft" contains the lettirs
+ u and — ii exactly once.
(The proof is identical to the proof of the Tremaux theorem.)
In other words, a path from a to />, if it exists, will be found by purely local
methods and without labelling the same arc more than twice. Note that if G is

PATHS, CENTRES AND DIAMETERS 59
an arborcscence rooted at a, the algorithm for new arcs and Trlmaux's
algorithm are equivalent.
Algorithm for a planar graph.
[fa chttin is sought between two vertices a and b in a pkinar graph G. and if
both a and b arc on the unbounded face, we have the maze problem. A traveller.
who tries to get out of the maze by following the passageways, can Find a
chain between a and b by the following rule:
At a junction, always take the passageways on the extr 'me right.
This iilgorithm always locates the exit to the labyrinth without traversing
any passageway mure than twice.
2. I'hc shortest path problem
The shortest path problem is the following:
Consider a graph G, and Jar each arc u, a number l{u) > 0. called the length
ofn. Find an elementary path from a to h that minimizes
Ktt) - I /(«)
am
Lxampi f. Find the shortest route on a map from city a to city />. To do
this, construct a graph G by representing each road between two localities on
the map by oppositely directed arcs in the graph: let the road's mileage cor-
correspond to the arc's length. Thus, this example reduces to a shortest path
problem. (Similarly, we might search for the fastest or most economical
journey.)
Shortest path algorithm (DantWg (I960])
Let the starting vertex be denoted by ax; we shall determine, for each vertex
■v. the length t(x) of the shortest path from n, to x by the following rules:
A) To begin with, let /(«,) - 0. Function / is therefore defined on the set
*x - {«»i }•
B) Suppose that on the fc-lh step, the function I has been defined on the
set Ak — { «j, fla flv }. For each vertex a, Ak, select a vertex b,s X — Ak
such that (a,, b,) e U and such that the length /(a,. /;,) is minimum. Find a
^rtcx aq e /<* such t hat
»(',) + /(«,. />,) - mm {i(as) + /(«,. b,)} .
Then, put
Kaq)

60
ORAPHS
It remains to show that t(hq) is the length of the shortest path to bt, and that
this path passes through aq. Assume that i{a,) is the length or the shortest
path to a,, Tor all a, Ak. All paths that leave Ak have a length > /(a,)
/(a«. 6<i) ■ '(*«)• To reach hq one must surely pass through Ak, since ax e Ah
Therefore i(bq) is the length of the shortest path from a to h.
Q.E.O.
Note that each time a vertex bq is selected, all the arcs with terminal end-
point bq may be deleted.
Remark. Suppose that the vertex b, that is associated with ate Ak can be
found without difficulty. Then the selection of the (k + I)-th vertex requires
only A; comparisons. The maximum number of comparisons needed for a
graph with n vertices is
I + 2 +
(n - I)- y\{n - I).
This bound can be improved if we search simultaneously for the length of
the shortest path from £>, to b.
Examim.i:. Consider the graph in Fig. 4.3. The number next to each arc u is
its length /(t/).
1. /(«) - 0. Compare ah - @ + I) and ac - @ + 2). Choose ah.
2. Mi) - I. Compare he - (I + 3),/i</ - (I + 3) and or - @ + 2). Choose
ac.
3. /(c) - 2. Compare he - (I + 3), hd - (I + 3) and </- B + 2). Choose
he, hd and cf.
4. /(p) - /(</) - /(/) * 4. Compare eg - D + 3). f/y - D + 3) and
fg - D + 2). Choose fg.
5. /(jr) - 6. Compare jr* - F + I) and fb - D + 4). Choose#6.
6. /(/>) - 7.
Thus a shortest path from a to b is flr/tffc.
The next theorem follows immediately from the above algorithm:
Theorem l.l/G — (X,U) Is a graph with root a. and with a length i(ti) > 0
for all ue V, then there exists in G a spanning tree II ■ {X, U) which is an

PATHS, CENTRES AND DIAMETERS 61
arborescence with root a, and such thai each path in II is a shortest path from
ainG.
Clearly, the different arcs chosen by the algorithm form an arborescence H
(since H has no circuits and is connected).
Analogous algorithm for symmetric graphs
For a symmetric graph, a simple algorithm to find a shortest path is to
represent the arcs or the graph by strings of the appropriate length and to let
the vertices be represented by knots that tie the arcs together. To find a
shortest path between knots a and b, simply pull knots a and b apart. The
taut strings between a and b will represent the shortest path.
3. Ctntrcs and radii of quasi-strongly conntctcd graphs
Consider a I-graph C ■ (X. V) and two vertices x and y of G. The directed
distance d(x.y) is defined to be the length of the shortest path from .v to y.
(If no such path exists, let d(x.y) - oo.) The associated number e(x) of a
vertex x is defined to be
e(x) - max d(x, y) .
A traveller at vertex x can reach any other vertex in e(x) or less steps.
A centre of C is defined to be a vertex x0 with the smallest associated
number. The associated number p(.v0) of vertex .v0 is called the radius of C,
and is denoted by p{G).
These concepts are important in telecommunications. A communication
network may be represented by a graph (not necessarily symmetric), and a
centre of the graph represents an optimal site for a transmitting station.
Proposition 1. The directed distances </(.v. ;•) satisfy
A) d(xtx)-0
B) d(x,y) + d(y, --)>*/(*, 2).
If the graph is symmetric, we hare also
C> '/(*,}') - d(y, x) .
The proof is immediate. If the graph is symmetric, ihc function d(x.y)
satisfies (I), B), and C) and is a distance in the topological sense.
Rccull(C"h. 3. § 3) that a "root "of a graph is a vertex x0 such that for each
vertex y, there is a path from x0 to y. Also, recall that a graph has a root if,
and only if. it is quasi-strongly connected.

62 ORAPKS
Cleurly, a centre has a finite associated number if. and only if the graph hat
a root, and therefore, if. and only if. the graph is quasi-strongly connected.
Henceforth, we shall assume that graph G is quasi-strongly connected.
Proposition 2. //'.v0 andy0 are two centres of graph G, then they both be/on
to the same strongly connected component.
The proof is obvious.
Theorem 2. lj'G Is a \-graph of order n. without hops, and such that
max dcix) - p > 1 .
then its radius />(C) satisfies
logp
If p(C) - + oo. the theorem is obvious. Suppose that />(G) < oo. Let x0 be
a centre of G. The number of vertices with u directed distance of I from x0
is < p. The number of vertices with a directed distance of 2 from x0 is <
Thus
n < 1 + p + p1 + - + f - " ^r—.
or
nip- I) + I <//+l,
and hence,
log(«yj - n + I) < (/» + I) logyj.
This gives the formula.
Q.E.D.
Lemma. Let G — (X. V) he a strongly connect •d graph of order n. and let
he a root ofG. Consider a spanning arhorescence II - (X. V) ofG with root
such thai each path of H is a shortest path of G. I el B denote the set of all
terminal rertices in arhorescenve II. Then
| B | e(a) > n - I .
Fqiiality holds if. and only if. II consists of\B\ paths of length t(a) startin
from a without common vertices iexc pt a).
The vertices of // (except«) can be placed on t(a) hori/ontiil lines such that
u vertex x with tl(a. x) - / is plnccd on the Mh horizontal line.

I'ATHS. CINTRI'S AND DIAMMTHS 63
lf/> e B, let X{a. b) denote the set of all vertices (except vertex a) in the path
of // th«t goes from a to b. Then,
n - 1 - I U *(«. « I < I I *(«. b) I < I /* I e(«).
|j>>a I hii
Q.E.D.
Theorem 3 (Goldberg A965]). If C is a strongly connected \-graph with n
vertices and m arcs, then
where [r]* denotes the smallest integer > r. For all in and all n. there exists a
strongly count cted I -graph with n vertices and m ores such that the above
equation holds with equality.
I. Let a be a centre of G. and let // be a spanning arborescence with root a as
defined in the lemma.
Since G is strongly connected, each pendant vertex be Bin arborescence //
is the initial endpoint of an arc in G — ff. Since the number of arcs in G — H
is in(G - //) - m(G) - m(H) - m - n + I, by Theorem (I, Ch. 3). we
have
\B\<m-n+l.
Then, from the lemma,
p{0) - ^ ^
This yields the required inequality.
2. We shall construct a strongly connected graph G with n vertices and m
arcs, with radius
' I. HI - N + M
This graph G will, in fact, be a rosace with centre a (sec Fig. 4.4). It con-
consists of m — n + 1 circuits (the "branches" of the rosace) having a common
vertex a. Let
n - 1 •
Thus, we can write:
n - 1 m (m - n + I) (q - \) + r; 0 < r < m - n + 1 .
distribute the n - 1 vertices (other than a) among the branches of the
r°sace by placing cither aora- 1 vertices on each branch. One branch will

64 OHAI'IIS
have at least q clifTcrcnt vertices other than a. since /• > 0. Hence. G is strongly
connected with n vertices, and />(G") - q.
Q.E.D.
I-Ir. 4.4. Rosace wiih ccnirc a.
ii - 15. m - IK,/) - 4.
We shall now study the properties of the centres of ii complete I-graph.
Theorem 4. If G - (X. T) is a complete l-grapfu each vertex x0 siuh that
| H.v0) - {.v0 } | - max | l\x) - { x } \
is a centre, and />(C7) < 2.
ir
max | /-(.v) - {.v } | - \X \ - 1 .
the theorem is true, and p(G) - I.
Otherwise, consider a vertex .v0 for which | /Xv0) — {Jfo} I is maximum.
Since d.\) > 2 for all .v. we huvc only to show that eix0) ■ 2.
Suppose that c(x0) > 2. Then there exists a vertex y x0 that cannot be
l-'lK. 4.5

PATHS. CENTHFS AND DIAMI'TI RS 65
reached from x0 by any path of length I or length 2. Since y I (.v0). then
x0 r(y)- Moreover, if r is a vertex of f(.v0) - {.v0 J, then yi !'(:), since,
otherwise, there exists a path of length 2 from x0 to y. Then z e l'(y). and
-iy)- Hence.
- {x0}
Since -v0 e /'(>■) - { .V }. the above inclusion is strict, and consequently.
| r(y) - { y } | > | n.v0) - { *o } | - '"ax | /(a) -{x}\.
xc X
which is clearly a contradiction. Q.E.I).
Theorem 5 (Maghout [l%2]). // G - (A\ V) is a complete I -graph with
radius 2. then for each y 6 X there exists a centre x0 such that (.v0. y) U.
Let y X- Since the radius of the graph is greater than I. there exists a
vertex *i »* y such that
Hence. (xt. y) U.
If x{ is a centre, the theorem follows. Otherwise, there exists a vertex
x2 Jfi. y. with (.v,. Afa) V und (y. x3) i V. Hence.
(xt,xi)eU. {xity)eV.
If x3 is a centre, the theorem follows; otherwise, there exists a vertex
*3 *i. *ai y such that
(x^xJiU. (Xl,xs)iU. ly.xJiU.
Hence
{Xs.xJeU. (Xi,Xl)eU. (Xi.y)eV.
If *3 is u centre, the theorem follows; otherwise, there exists a vertex .v«.
etc.
At least one vertex xk kKutcd by this procedure is a centre. Otherwise.
Therefore a centre of the subgraph generated by X - {y) is also a centre of
graph G. and by Theorem 4. a centre would have been located during the
Procedure.
Thus, this vertex xk is the required centre. Q.E.D.
Corollary. A complete \-graph G with radius 2 has at least 3 centres.
Let G' be a complete, anti-symmetric graph obtained from G" by removing
sonic of its arcs. This new graph C has a centre. yx und. by Theorem 5. there

66 URAPHS
exists another centre y3 with (.va. ;•,)£ V. Also, there cxisis a centre yA with
(>»>"a)e f- Since ihc graph 0" is anti-symmetric.
y> r,.
thus. G has ut least three distinct centres »-,. .r2 and jv
Q.E.U.
4. Dlamrtvr (tf a stronnly connected uraph
The diameter 6{G) of a graph G is the maximum of the directed distances,
i.e.:
ti[G) « max </(x. >•) .
jr • y
m i.p-A.S 4. hi-y,/>-2,<5 4, m 6,/>-2,<5 4.
V
Kosncc
Rosace
m — 6,
— 3. H 4.
Rouicc
mi » 6,/J «■ .1, E 4.
-T,P-2.fi 4.
Hll. 4.6
- 20./> - 1.6 -

PATHS. CTNTRrS AND DIAMITrRS 67
The diameter is finite if. and only if. G is strongly connected. In this section,
xvc slv.ill assume that G is always strongly connected. Also, without loss of
K(.ncrality. we ni.iy assume in this section ih.it G is .1 1-graph without loops.
f-'XAMi'i.!:. In Fig. 4.6. each graph has S vertices and is strongly connected.
The number of arcs is denoted by in, the radius by />, and the diameter by 5,
A circle is drawn around each centre. A square is drawn around each vertex
.vsiich that i[x) - MG).
Note that for n « 5. the equation 6 + m > 9 is always satisfied. Only the
complete symmetric graph has a diameter equal to I.
Since the graph is finite, then clearly. A < j. and if the graph has a centre.
fi % p. In a graph representing the avenues of communication between the
various members of an organization, the diameter 6 represents the maximum
number of times that a message must he relayed before it reaches its destina-
destination.
The problem of constructing strongly connected graphs with n vertices and
in arcs whose diameter is as large as possible (or 11s small as possible) was
considered by liralton [1955]. The problem of maximum diameter graphs has
been solved by Ghouila-Mouri [I960], and the problem of minimum diameter
graphs has been solved by Goldberg A966],
Before proceeding, note that if 6' is a strongly connected graph without
loops and with n vertices and m arcs, the numbers n and m cannot be chosen
arbitrarily: If n > I. G has at least one cycle, hence, the cyclomatic number
i(G) « m - n + I > I. and consequently, we have
m > n.
From Theorem (9. Ch. 3). we know that equality holds only if G is an
elementary circuit.
Furthermore, since the number of arcs in G cannot be greater than the
number of arcs in a symmetric complete graph with n vertices, we have
m < »(h - I).
Ihcorcm 6 (Goldberg [1966]). l/Gis a strongly connected \-graph without
loops (uul with n ivrtices and m arcs, and if G is not an elementary circuit, then
m - n +
t'unhemi ire. this is the best possible result.
'f G is not an elementary circuit, then, from Theorem (9. Ch. 3).
m- n + I m v{G) > 2.

68 ORAl'HS
Consider a strongly connected graph G with cyclomatic number v(G) ' *
we shall show first th.it
v(G)<5(C)>2(«- I).
A trail of G is defined to be any elementary path >i - |.v0. xlt .... x,] sue
that for / - 0. I s — I. only one arc of G is incident out of *,. Let jr
denote the initial vertex of a longest trail of G. say:
fi0 - [.v0. ...,*,].
Let .v be the length of /<0. Construct an arborcsccncc // rooted at x0 as defined
in the lemma to Theorem 3. Clearly. // begins with path >i0. and its first node
after x0 is x,. Let B denote the set of terminal vertices of arborcsccncc //.
1. We shall show that | B \ + I < vF").
The number of arcs in G - // is v(C). Since each vertex be Bis the initial
endpoint of at least one arc of G (because G is strongly connected), we have
WO > | B |. If r(O - I # |. then each vertex of B is the initial endpoint
of only one arc of G, and there exists a vertex bxe B such that (/»,. xo)s V
Clearly, since | B \ - v(C7) > 2. the vertex bx is not on fi0 ■ [-vo< -^i x,),
and [/>i. .v0. Xi ,v»l would be a trail of length .t + I. This contradicts the
maximality of >/0. Hence, v(C) > \ B\.
2. We shall show th.it s > i f>(G) implies h - I < \ S(G) v(C).
Let v(G) - v, and let 6(G) - d If we denote the ussociated number of
vertex * in the arborcsccncc // by fw(.v). then, from the lemma to Theorem 3,
n - 1 < .« + e,,(xt) \B\*s + C-s)\B\m
\B\-t{\B\- l)<3\B\-?(\Bl-l)m
3. We shall show that s < ' implies
y
Let
■" i m { xo* xi • •••» A'« / •
X2-{xlxeX, e,,(x) < s - I}.
Note that Xa can be empty (if s - 0). and that xot Xt. If A'a »* 0, note
that //Aa is the union of arborcscenccs; denote by Bs the set of all their

PATHS. CENTRES AND DIAMF.TEKS
69
terminal vertices. The number of maximal paths in Hx> is | B3 |, and the num-
her of vertices encountered by such a path /<3 is n(pa) < S - 2 s (because
6 ? /(/'I*o. b]) > s + n(na) + s). Hence,
I_ct fH(z) be the set of all the descendants of z in the arborescence //,
and let '(•) denote the number of arcs in C - H with initial endpoint in
We shall show that zoe£j implies t(z) > 2. If. for u vertex zoe B3,
r(z0) ■ '• l^cn zo 's l^c initial endpoint of only one maximal path ^[z0. />]
in //; furthermore only one arc of G - His incident out of the set of vertices
encountered by >([z0. b). This arc is of the form (/>. r) with r e [z0, />] (because,
otherwise, C would contain u trail of length s + 1).
Hence, there is no path from z0 to x0, and the graph is not strongly con-
connected, as there is no path from z0 to x0, which contradicts the hypothesis.
Thus, /(z) > 2 for zeBa. Therefore
l*Oi
Finally, we have
Hence.
Mg. 4.7. Strongly connected graphs wiih
, . r «fli) r r to
«-6 and J-^j [

70 OK AMIS
4. Purts 2 and 3 of this proof have shown that ifv(C) > 2, then
rS(G) v(t7) > 2 0i- I),
and thus.
Since it is not difficult to construct a rosace G Tor which equality holds (see
Fig. 4.7). the proof is complete
Q.E.D.
We shall now discuss the problem of maximum possible diumeter.
Lemma 1. A necessary and sufficient condition that a strongly connecte
\-graph G - (X. U) hare diameter 6(G) > p is that X am be partitioned hit
p + I classes Xa, Xx Xf with:
*€*,. yeX,. (x,y)eU ~ ./< / + I .
Sufficiency. IT such a partition exists, a path from ,v0 6 Xn to x, e X,, is of
length > p. and. thus. MG) > p.
Necessity. If S{G) > /j. consider two vertices a and /> with </(«. b) "*• p.
The sets:
X, - {xIx6 X, d(a. x) - /} (/ - 0. I /»-!),
*„ - { x / * 6 X. (Ha. x) > p}
are non-empty and form the required partition.
Q.E.D.
Lemma 2. Let p > 2. There exists a strongly connected \-graph with n irr»
tices and m arcs whose rertex set can be partitioned into p + I classes Xot
Xi,..., Xf as in Lemma /. if, and only if,
..2
<D m <y- n + Mn.p),
B) m > n ,
C) p < n - I .
Let A" be a set of cardinality n. nnd let (XOt Xx Xr) be a partition of X
in p + I > 3 classes. Let | A", | - «,. Clearly, /> < n - I. We shall show that
for each w that satisfies (I) and B). there exists a strongly connected 1-grapli
on X with m arcs that has (Xo, Xx Xp) as a partition. First, construct

PATHS. CENTRES AND DIAMETERS 71
strongly connected graph by placing, between these n vertices, n arcs with the
property of Lemma I: for instance, take an clcmcntnry circuit that first
encounters nil the vertices of Xo, then nil the vertices of A",, etc.
Now successively add to this grnph m - n new arcs so that the property
of Lemma I is preserved.
CIcnrly the mnximum possible value for m equals:
1) + «o "i +
+ n, n0 + m,(m, - I) + Hi m2 +
For
Up
nnd
+ n,i
--,
integers n > 3
/(m0, m
- 2, then
/(
'»O + M,M,
H + - + J
nnd /? with
1 Mp) ™
Mo* Mi r /tg)
,1.
+
■*'
2
1
-
•••
<
+ np np.
n\ + n0 i
p < « -
JcO
•o +
1
2
mnx
n, > 1
(M0 + M,
I- "a
1 "
-i
«i
l,
(«c
A'
+
I
1
+ np(np- 1) -■
+ - + 'i,-i ",.
let
,'„+•••+'-,.,',„)
V«i »,)■
•
Let /> > 2: let (»0."i '',.) he a (/> + l)-tuple that maximizes /. If
"i > 1 and n, > I with / - / > 1. then the value of/can be increased cither by
taking «; _ M| | nntj ,,j _ n> + \_ or by taking «,' - «, + 1 and nj - n, - I.
since
Mo «, ± I n, T 1 ff,) »
■/("o nr) + I + («,., + n, + «,.,) T ('«,-! + n, + n,,i).
It follows that there exists an index k. with I < k < /> - 2 such that
/ A "
/ A- +

72 ORAPIH
Therefore,
n _ I h? -1. h? .
P-3
- 3 + - (nk
By replacing m* + «„♦, + 1 by «-(/»- 2), we obtain
«a
Since this equality is also true for p - 2, the theorem follows.
Q.E.D.
Theorem 7 (Ghouila-Houri [I960]). The maximum value for the diameter o
a strongly connected loopless \-graph with n vertices and m arcs is
n" + n - 2
- n - 1 if n < m <
(m, n)
., h8 + « - 2
if < m < m(« - 1).
Let C be a strongly connected loopless graph with n vertices and m arcs;
then m > n because each vertex is the initial endpoint of at least one arc.
Furthermore, m < n(n - I), since the graph cannot have more arcs than a
complete symmetric 1-graph, If in - n(n — I), the proof is immediate;
therefore, we may assume m < n(n - I) and m > n.
1 f conditions (I) and C) of Lemma 2 are fulfilled, then from Lemmas I and
2 there exists a strongly connected l-graph with S(G) > p. Conversely, if
conditions (I) and C) are not satisfied, then no such graph exists. Thus, the
greatest possible value for the diameter is the greatest value <p(m. n) for an
integer p satisfying (I) and C).
Inequality (I) is equivalent to
I. . („
+ „ - 2 - ».) > 0 .
Then, by elementary calculus, we obtain the above value </>(»'. >') for p.
Q.E.D.
For a simple graph G — (X, £), we define the undirected radius p*(G) as

PATHS, CENTHI'S AND DIAMETERS 73
•he radius p(G*) of the graph G* - (X, U) obtained from G by replacing
each edge in G by two oppositely directed arcs. Similarly, we define the
undirected diameter S*{G) of a simple graph G' by
5*{G) - &(G*).
The following theorem is due to Camillc Jordan A869].
Theorem 8. ifG is a tree, and if6*(G) is even, then G has a unique centre,
ami all 'he elementary chains of maximum length pass through it: furthermore.
JfS*(G) is odd. then G has exactly two centres (which are adjacent vertices)
and all the elementary chains of maximum length pass through them; further-
furthermore.
AC) - 5 (S'(G) + 1)
The theorem is obvious for trees of order < 3. Let n > 3, and suppose that
the theorem is true for trees of order < n: we shall show that the theorem is
true for a tree G - (X. E) of order n.
Let B lie the set of pendant vertices of tree G. From Theorem B, Ch. 3). we
have | B \ > 2, and hence, the subgraph GX.B is a tree of order < n. Clearly,
I>*(GX-B) - f>*{G) - 1 .
Each centre of GX.B is a centre of G, and vice versa. Each elementary chain
of rruixinuiin length in Gx-b induces in graph G an elementary chain of
length 6*[G). Since the theorem is true for CX.B. by the induction hypothesis,
the result follows.
Q.E.D.
Rtmark. The two equalities of Theorem 8 can be summarized by:
/(O-[-2(<5*(G)+ 1)],
where (/■] denotes the greatest integer < r.
For simple graphs, several results regarding f>*(G) and 6*{G) have been
obtained: Vizing A967] has determined the maximum number of edges in a
simple graph with p*(G) - r. Murty [1968) has determined the minimum
number of edges in a simple graph with S*{G) < k such that the diameter
'cmuins < / after the elimination of any s vertices. The diumetcr of a planar
3-connectcd graph has been studied by M. Balinski A966]. D. Grilnbawn
l'9<7] Bollobas A968]. Bosdk. Kotzig. Zmtm A968].

74 ORAPHS
5. Counting paths
This section presents results about the number of different paths between
each pair of vertices x, y in a graph G — {X, U). It will be necessary to use
matrix products as defined in linear algebra. The principal result for path
counting is:
Theorems Consider two graphs G - (X, U)andH - (X, V) with the sam
vertex set. and let A - ({a\))andB - ((/>})) denote respectively their associate
matrices. The matrix product AB corresponds to a graph G. H with vert x
set X, and an arc from x to y for each distinct path from x to y composed o
an arc of U followed by an arc of V. The graph G. H is called the composition
product of G and H.
The number of distinct paths of the form [x,,xk,x,) with (xitxk)e r
(xk, x,) e V, equals at, b). Thus, the total number of different paths from *
to x, formed from an arc of V followed by an arc of V is
where < a', b, > denotes the scalar product of the row vector a1 and the
column vector b.,. which is also the general coefficient of the matrix A . B.
Q.F.D.
Corollary 1. If G is a graph and A is its associated matrix, the general co-
coefficient p\ of the matrix P - Ak {the product of A with itself k times) equals the
number of distinct paths of length k from x, to x, in G.
The theorem is true for k - I. Let A: > I, and suppose the theorem is true
for the power k - I. Then it is also true for the power k, since A" - A(A" ')
shows the number of paths of length I + (k - I) - k from x, to x, (by
Theorem 9).
Q.E.D.
Corollary 2. A graph G possesses a path of length k if, and only if. A" 0.
G possesses no circuits If, and only If, Ak - Ofor k sufficiently large.
The proof follows immediiitcly from Corollary I.
Application. The power index of a participant in a partial tournament.
Let G - (X, V) be a I-graph of a partial tournament with n participants.
Let arc (.v. >■) e V if participant x has beaten or had a draw with participant
It is tempting to choose as the winner the most dominuting participant. i.0>
the vertex x whose outer demi-degree is as large as possible.

PATHS, CENTRFS AND DIAMETERS 75
However, there is a good chance that this participant x has beaten a large
number of very weak participants, and would lose to u participant y who has
beaten only a few very strong players. IT p{(k) is the general coefficient
of matrix A". i.e., the number of paths of length k from x to x, then a better
estimation of the power of pnrticipnnt .v, is the sum
Ak) - p\(k) + pl2(k) + - + pln(k).
Consequently, the power index of participant xt can be defined as
n' - lim .
p'(k)
*-« p\k) + p*(k) + - + p"(k)
Note that by virtue of the Perron-Frobenius Theorem, this limit always
exists.
EXERCISES
1. Lei 8*(C) - iuf>fHx,y) be (he diameter of a simple connected graph G. Show ihat
the following conditions arc equivalent
(I) Any iwo vcriices of G are connected by ai moil one elementary chain of length
B) G has no cycles of length < 2 8*(G).
C) F.i(hcrihcrccxls(nocytlcsinC,onhclcngihofihcshorics(cycleorGi»2
2. Given two integers A and 8, the number n of vertices of a simple griiph regular of
degree h wlih diameter * satisfies
t
A simple graph with n veriices, regular of degree h. wilh diimieter 8. for which the
equality holds is culled a Moore graph of type (//. 8).
Show thai the Petersen graph (Fig. 10.11) is a Moore graph of type C, 2). Show thai
■he Moore graphs of type {h. I) are the (h + D-cliques. Show that ihe cycles wlihout
chords of length 2 8+1 itrc Moore graphs of type B, &).
Show thai I here exists a Moore graph of lypc G. 2) und thai it is unique (A. Hoffman,
K- Singleton [I960)). (Hoffman and Singleion have characicrized I he Moore graphs of
■ypes (h, 2) and {h. 3) with the exception of type E7. 2) for which no exumple is known.)
}• Show ihm a aimplc connecicd graph G saiisfies the equivalent propcriies in txcrcisc I
"• »nd only if. it is ei(hcr a iree or a Moore graph.
(J. Bosak. A. Koi/ig, S. Znam [1968])
■ Show that for a simple connecicd graph, two clemeniary chains of maximum length
chn"yS ''aVc conlnu)n veriices. Also *how ihai if k is I he length of a longesi clemeniary
a|n, ihen each vertex Is the iniiial endpoint of an elementary chain of lengih
* I Wo + I)].
' Sl'0^ 'hai a simple gruph G of order // > I which is feomorphic lo iis complcmcniury
*r»Ph G has a dinmcier cquiil lo 2 or i. (II. Sachs [IV62])

CHAPTER 5
Flow Problems
1. The maximum flow problem
Consider a graph G with arcs denoted by I, 2 m, and consider a set o
numbers bu ba,..., bm, cu ca, .... cm in Z such that
- co < 6, < c, < +oo.
A flow in G is defined as a vector <p - {ipy, v>a <pn) Zm such that:
A) <p, 6 Z for / - 1. 2,.... m. (The integer <p, is called an arc flow, and may
be regarded as the number of vehicles travelling through arc / along its dir
lion if tpt > 0 or against its direction if <p, < 0.)
B) For each vertex .v, the sum of the arc flows entering x equals the su
of the arc flows leaving .v, i.e.,
I <Pi - I <Pj (xeX).
In other words, there is a conservation of arc flows at each vertex (Kirch
hofT's Law).
We shall study the following problems:
Ihc compatible (low problem. Given a graph G with an interval [b\.C,
associated with each arc i,fimi a flow <p such that
b,i<fi,< c, (/- 1.2 to).
fi is called the capacity of arc ;', and represents the maximum number
vehicles that can use the arc / along its direction.
The maximum flow problem. Given a graph G with an interval [l>,, c,) assoct
ated with each arc I, find a flow tp such that
A) /;, < tfi, < c, (/- 1,2 wi).
B) the arc flow </), is as large as possible.
Note that, For both of these problems, we may assume without loss
generality that G is a I-graph.
76

FLOW PROBLEMS 77
The maximum flow problem occurs frequently with the following additional
conditions:
A) b, - 0 for/- 1.2 ro.
B) ct > 0 for all /, and cx - + oo
C) arc I is an arc joining a vertex b, called the sink, to a vertex a, called the
source, where
a>-(a) - A,0,0 0),
w+(ft)-A,0,0 0).
D) G is an anti-symmetric I-graph.
In other words, only arc / - I enters vertex a, and only arc / - I leaves
vertex b. This arc (b, a), generally omitted in the illustrations, is called the
return arc. (This arc has no function other than the maintenance of the con-
conservation of flow at vertices a and b.) The most important flow problem is to
find the maximum number of vehicles that can be sent from a to b without
violating the arc capacities.
A graph G with a capacity c, associated with each arc / and which satis-
satisfies the conditions (I), B), C), D) is called a transportation network and is
often denoted by R - {X, U, c(u)).
Below are some examples that reduce to maximum flow problems in a
transportation network.
Example I. Maritime traffic. Let the seaports au a a,, bu b2,..., bt,
be represented by vertices, and suppose that bananas arc ready fur shipment
at ports a,, a a, to ports blt b , bv Let s, denote the quantity avail-
available at a,, and let d, denote the quantity demanded at b,.
Shipping routes can be represented by arcs of the form (o,, b,) with a capa-
capacity equal to the shipping capacity between the two seaports. Is it possible to
Mtisfy all the demands? How should the bananas be shipped? To answer
these questions, create a source a, and join a to each a, by an arc with capacity
r(o. a,) - s,.
Next, create a sink b. and join each vertex b, to b by an arc with capacity
c(bJtb)-dj.
A maximum flow for this transportation network yields the number of
a"anas to ship along each route in order to satisfy all demands, if this is
Possible.
Example 2. 77tc> battle of the Marne. The towns aih aa an each have

78
ORAPHS
motor cars to be sent to town b. IT there is a relay route from town a, to town
a,, let c,, denote the number of motor cars that can leave ax for a, each time
period. Let i, denote the traverse time from a, to a,, let s, denote the number
A E)
Sink
Fig. 5.1
of motor cars initially available at a,, and let fi denote the number of motor
cars that can park at a', simultaneously. Mow can we direct the transport
of these motor cars so that as many as possible arrive at h within T tim
periods? This type of problem has been studied by R. lulkcrson under the
name of "dynamic networks". The problem can be reduced to a maximum
flow problem in a transportation network with vertices a,{i), where / «■ I*
2 //.and i - 0, I T.
Vertex a,(i) and vertex a,(i + I) arc joined by an arc of capacity c,. If there
is a route from <v, to a,, it is represented by an arc from a,(t) to a,{t + '
with a capacity of c,,. Add a source a. sink />. arcs (a. r/,@)) with capacity
and arcs (/>(r). h) with capacity oo. The maximum flow in this transportation]
network determines the optini.il routes.

fLOW PROBI IMS
79
3. rite selection of representatives. A set X of residents belong
to various clubs C,.C C\ (which are not necessarily disjoint subsets
of X) and to various political parties P,.!1*,.. P, (which arc disjoint
subsets of X). Each club must choose one of its members to represent
it. and no person can represent more than one club, no matter how muny
clubs he belongs to. How should one choose a system of distinct representa-
representatives A - {a,, a-i a,}, such that the numbers of representatives belonging
to each party P, satisfies
A solution to this problem is given by a maximum flow in the transportation
network shown in l-'ig. 5.2 (for a — 4 and r - 3).
X
Sink
Auxiliary vertex
Mr. 5.2
The arc capacities arc marked in parenthesis. It is left to the reader to verify
a maximum flow that saturates all source arcs determines a set of distinct
rcPrescntatives as required

80 GRAPHS
General maximum (low algorithm (when a compatible (low is known) (Ford and
Fulkcrson [1956]).
Consider an anti-symmetric I-graph G with arc I - (b, a). Suppose that
compatible flow <p satisfying bt < <pt < <*■ is known. We shall augment suc-
successively the value of the arc flow 47, by the following labelling procedure:
Rule I. Label vertex a. the terminal endpoint of the tire I. with the inde
+ 1.
Rule 2. If .v is labelled and y is unlnbcllcd. label y with the index + k if
(x,y) is arc k, and if
tp(x, y) < <(.v, v).
Rule 3. If v is labelled and y is unlabcllcd, label y with the index - A- i
iy.x) is arc k, and if
9(y. x) > b(y, x).
If sink b is labelled by this procedure, we shall show that the arc flow tpx can
be augmented, i.e., we shall construct a new flow <p' such that <p\ > v>i ■
Let
H - [a, a,,a2 ak, b]
be a chain from a to b in which each vertex a, + 1 has been labelled from its
predecessor a,.
(I) If the edge [a,, a,*i] is directed from a, to a,tl, then we have
Then, put
B) If the edge [a,, a, n] is directed from altl to a,, then we have
Then, put
(p'(aj+1, aj) - <p(a)t,, a}) - I :
C) For arc I - (b, a), put
V>'(/>. fl) - <p(b. a) + 1 - <Pt +
D) For all other arcs, put
(p'(x, y) - v>(*. y) .

FlOW PROBLEMS 81
In this way a new Mow <p' is constructed. Flow 9' is compatible because only
the flow around a cycle ji' - p + [ft, a] has been changed, and at each vertex
of this cycle, the conservation of flow has been maintained. In fact, the new
flow can be expressed as the vectorial sum
9' - «P + M' •
It remains to show that if the algorithm cannot label sink ft, then the arc
Jlow*P\ is maximum.
Lemma 1. In graph G ■ (A". U), let A <=■ X be a set with as A andb$ A.
Then, for each flow <p. such that ft, < tpt < c,,
<P\ < I c, - £ b, .
it«°H> 1*1
Since the algebraic sum of the Mow entering set A equals the algebraic sum
of the flow leaving set A, it follows that
if I liu'(A)
Hence.
■ I ft- I ft< I
Itw'lAl 1*1 tto'i
'{A) (*l
Q.E.D.
I emina 2. If the Ford and I'ulkermn algorithm cannot label sink ft, then tf>x is
maximum.
Let A denote the set of vertices labelled by the algorithm. Clearly, a s A and
. Since no more labelling is possible, an arc (x. y) with .v e A and y$ A
satisfies <p(x,y) - c(x,y), and an arc (y,x) withy^ A and xsA satisfies
• •*) ■ ft(y, -v). Thus,
From Lemma I, it follows that tpt is maximum.
Q.E.D.
Theorem 1. (Ford. Fulkcrson [1957]). In a graph G with numbers h,, c,
mere — 00 < ft, < r, < +00, r/re maximum value of a compatible flow in arc
1 is
max 9, - min / £ c, - £ ft,\
The proof follows from Lemma I and 2.

82
GRAPHS
In a transportation network R ■ (.V, U,c(u)), a cut between a und b i
defined to be u set of arcs of the form ta'iA) with tie A and hi A. Fh
capacity if a cut is defined to be the sum of the capacities of its arcs, i.e.
Maximum How theorem (Ford, Fulkcrson [1957)). In a transportation n t
work, the maximum value of the arc flow tpi equals the minimum capacity c
cut.
Example. Consider the transport network in Fig. 5.3.
Via. 5.3
The return arc (/>, a) has been deleted to simplify the figure. The capacity o
each source arc und sink arc is 2. The capucity of each intermediate urc (arc
that iire neither source aircs nor sink aircs) is I. A compatible flow</> is easily
found, and the sat united arcs arc indicated by heavy print. Its arc flows ure
tp{a. <•) - tp(a, <l) m 2 ,
Via. e) - tp(e. It) - </>(</, f) - tpitL a) -
- <Picf) - ipic. S) - <*»(/'. f>) - I
This flow is nia.ximal'm the sense that there urc no paths from a lob composed
entirely of unsaturated arcs. However, the flow is not maximum because the
labelling algorithm can be used to construct a flow <p' with <px - <p\ 4- I. For
example, the vertices can be labelled us follows:
*<+»): n+d): Ri+ex); c(-cf): ill- tig) ■
ch); h(+hb)

I LOW PRODUMS 83
This dclcrinincs the chain:
fi[a, b] ■ + ae + ef- cf + ch + hb.
Let
tp'(a. e) - tp(a. e) + 1
»'(*./> - Vie./) + I
«>V./) - Vic./) - I, etc.
The reader can easily verify that the new flow <p' is maximum.
Note th.it for a network of this type, a maximum flow can be obtained
directly by using the following rule: during the sequential flow augmentation,
send an additional unit of (low toward the vertex that hats the greatest capa-
capacity to receive flow. When this flow has been constructed, it will be maxi-
maximum. However, this simplification is possible only for very special networks.
In fact, the Maximum (-'low Theorem simplifies even further when the
network R is bfpanii . i.e., the vertices form four disjoint sets, X. >', {«},{&}
and the arcs arc of the following types:
type /. (x, y) with x<- X.ye Y,
type 2. (a. .v) with xe X.
type 3. (>', b) with y e Y,
type 4. (b, a), the return arc, denoted by 1.
To simplify, let
ciy.b) ifyeY
0 otherwise,
lKy) is called the demand at vertex y. If B <= Y, then the demttml of set B is
defined to be
<«>■>-{;
d(B) - X d(y).
If B c Y, let F\B) denote the maximum quantity of flow that can be sent
'« B, i.e.. the maximum flow for a network R' obtained from R by changing
c capacities in the following way:
c'iy.b) - + oo ifysB
c'iy.b) - 0 if ye Y- B
c'ix, y) m dx. y) for all other arcs (x, y).
e now have the notation needed to state:
Theorem 2. In a bipartite transportation network R - (A\ Y. V). the maxl-
«" value o/a compatlbi Jlow In arc I Is

84 GRAPHS
<px - d{Y) + min [F(B) - d(B)] .
uc r
1. Consider a set Be Y. and construct from R the network R' as describ
above. The Ford and Fulkcrson Theorem states that
F(B) - min c'[w"(S)] .
We may restrict our attention to sets S containing B because, othcrwi
r'[a)~E)] ■ + oo. Without changing in the right side of the above equation
S can be replaced by 5 -{/'}. Since we are only interested in the minimum
we may also restrict our attention to sets S of the form S ■ A u B, whei
/(cl Thus
F(B) - min c'[u>~(A u B)] - min c[(o'(A vj B)l.
Ac X Ac X
2. If P and Q arc two disjoint sets of vertices, in order to simplify, let
f(P. fi>- I c(p.q).
per
Consider a.set S with b e S, a £ 5. Let
SnX - A.
SnY - B.
Then,
c[ftT(S)J - c(a, A) + f(X - A. B) + c(y - B, h)
- c(w"(/l vj B)) + d(Y - B) - c(a>~(/i u B)) ■ d(B) +
From the Ford and Fulkcrson theorem, the maximum value of a flow in arc I
equals
</>i - min c[w~(S)l - min min [d{Y) + c{to~(A u B)) - d(B)]
Sab Uc t AC X
Sin
- d{Y) + min (F{B) - d(B)) .
Bet
This proves the theorem.
Q.E.D-
Corollary (Gale [1958]). A bipartite transportation network R ■ (X, Y, V
has a compatible flow that saturates all the sink arcs if, and only if,
F(B)>d(B) (BcY).

FLOW PUOIII EMS 85
min (F(B) - <t(B)) - 0 .
Jlc Y
In fact, this condition is equivalent to
■ it(U)) - 0 .
Q.E.D.
\lgorithni for planar networks
If the transportation network R is planar, the following algorithm m.iy be
used:
1. Place arc {h,a) horizontally in the plane, with vertex a at the left
with vertex/> at the right. Draw the graph above arc (/>. a) such that the arcs
Jo not cross one another. In this topological planar graph, find a superior
path )>' from a to h by nlwaiys choosing the left-most arc (and hack tracing
when nn impasse occurs).
2. Reduce the capacity of each arc in the superior path ft1 from a to h
by the amount
Ci ■ min c,.
leu'
fcliminatc from the graph all arcs with zero capacity.
3. Similarly determine the superior path ft3 in the new network. Reduce the
capacity of each of its arcs by the amount
e2 ■ niiru'i.
4. Repeat this process until there arc no more paths from a to b. The
required flow is obtained by sending r* units along each path /i\
To validate this algorithm, it is first necessary to establish the following
lemma.
I-crania. Let A be a set of rvrtices such that a A, b A and such that
W'M) is u minimal rut. The superior path fil encounters tu '(A) exactly once.
Clearly, path ft1 encounters the cut at*(A) at least once. Suppose that the
Path encounters the cut at two arcs, say / and./, in that order. We shall show
thut tnce exists a path ft from a to h that meets to*(A) only at arc /. Colour
*rc ' red. the other arcs of <a'(A) green, and colour all other arcs black.
*D) is a minimal cut, arc I is not contained in any green and bluck
in which all black arcs are similarly directed. By the Arc Colouring
rna, (Ch. 2), it follows that arc I belongs to a red and black cycle with all
k arcs having the same direction.

86
GRAPHS
Thus, this cycle is a circuit, and it induces the required path ft.
Similarly, there exists a path v from a to b that meets <o'(A) only nt arc .
Il«. 5.3
Since ft1 is the superior path, the portion of path ft following arc / and th
portion of v preceding arc j have a common vertex .v. This implies tha
x A and x <f A, which is a contradiction.
Q.E.D.
Proposition. I'he low «p obtained by the Algorithm for planar networks
maximum.
Let An denote the set of vertices that can be reached by a path from
Then
aeA0, b 4A0.
Consequently, w'(A0) is a cut. Let to*(A) be a minimal cut contained ii
w'(A0). l7rom the lemma, the A-lh step reduces the ciipacity of the cut w4(
by exactly e*, and the procedure terminates with a zero cut capacity. Thus,
Q.E.D.
* tta'(A)
By I cinma I to Theorem I, flow 9 is maximum.
2. The compatible flow problem
Clearly, there always exists a compatible flow in a transportation network
since the flow <p - 0 is compatible. However, there may not always exist

FLOW PROBLEMS 87
flow compatible with the constraints b, < <p, < r,. First, we have:
fl. A necessary condition for the existence of a compatible flow is that
for each cocycic w(A),
I ct- I fc, J>0.
Ita'lA) Iiii-M)
Clearly, if a compatible flow «p exists, then
£ c, - £ fc,.
'(A) ttv)-(A)
Q.E.D.
0 - I (P, - I (p, < £ c, - £ fc,.
ita'(A) ttm-(A) I* a'(A) ttv)-(A)
The sufficiency of this condition will be demonstrated later.
Compatible How algorithm (J. C. llcrz [1967]). Let G - (X, U) be a graph
with numbers />, and r, associated with each arc /, By using an iterative
procedure similar to the Ford and Fulkcrson algorithm, we shall construct a
compatible flow starting with any non-compatible flow <p.
Let the distance of ip, from the interral [bx, c,] be defined to equal
if VieCVr,],
if <p, < b,,
if (p,>c,.
The algorithm successively reduces the value of
I
lff/(<p) » 0, tlow <p is compatible and the procedure stops.
M'f/(<p) > 0, there exists an are i with </,(<p,) > 0. Suppose for example that
> 0. and that </>, < bt. Then. scc|iicntially. label the vertices accord-
according to the following rules:
; |. Lubcl vertex a, the terminal endpoint of arc I with the index +1.
2. If jc is labelled and y is not labelled, label y with the index j if
>') is arc J and if (p, < c,.
3. |f .v is labelled and y is not labelled, label y with the index —j if
OJ.-v)isarc./andit>, > b,.
H vertex b, the initial endpoint of arc I. is labelled, then a new flow <p such
"at d(y') < ^,p) can j,t. constructed by using the method of the Ford and

88 ORAPIU
Fulkcrson algorithm. If the initial endpoint of arc I cannot be labelled, the
the set A of labelled vertices satisfy aeA and b<$A. Since <px < blt it follow
that
0 - £ (Pi - £ Vi > £ c, - £ b,.
1«u*M) fa-(A) Ku'Ml lium<M
From the lemma, no compatible flow exists, and the problem has no
solution.
This algorithm establishes the following result:
Compatible flow theorem (Hoffman [1958]). For a graph G with arc numbcrt
bt andc, such thai — oo < b, < c, < + co for ail i, a necessary and sufficient
condition that there exists a flow <p with b, < <p, < c, for </// / is that, for each
set A c x,
I c, - I *,>0.
lltfiA) Itu-(A)
The necessity of the condition follows from the lemma.
The condition is sufficient because each flow 9 with d(<p) > 0 can be im-
improved by using the preceding algorithm, until we obtain a flow 9' witk
d(<p) - 0.
Q.E.D.
The following algorithm is in general more efficient.
Second compatible flow algorithm
Let G be a graph with vertices xuxa xn and with arc numbers bit ct.
The following rules construct a flow <p in G such that
b,<V, <c, (i - 1.2 m).
Ruie I. Construct a transportation network R' from G by adding
source a, a sink b. a return arc (/>, a), and the various source and sink arcs.
Rule 2. \tb(x.y) > 0. let the capacity of arc (x,y) in network R' equal
c'(x. )•) - c(x, y) - b(x, y) ,
and create a sink arc (.v, b) with capacity
c'(x, b) - b(x, y).
Also, create a source arc (a,y) with capacity
c>(a.y)-b(x,y).
Rule 3. If b(x.y) < 0. let the capacity of arc (.v. y) in network R' equal

FLOW PROBLEMS 89
c'(x, y)- c(x,y) - b(x,y),
and create a source arc (a, x) with capacity
c'(fl,x)- -b(x,y).
Also, create a sink arc (y, b) with capacity
c'(y', b) - - b{x, y) '.
We shall now show that a compatible flow exists in G if, and only If, in
transportation network R', there exists a maximum flow <p' that saturates all
source arcs and sink arcs.
If we let <p(x,y) ■ <p'(x,y) + b(x, y). then clearly 9 is a flow in C because
for each vertex x in G,
»i » 0
p'j + fc.) - £ (¥>} + bj) - 0 .
Flow 9 is compatible in G, because
0 < <p\ ^ Cj — bi (/ ■ 1, 21..., mi) ,
and consequently,
/>, < p, <c, (/- 1,2 m)
Conversely, it is easily seen that each compatible flow 9 in G corresponds
in R' to a flow 9' that saturates all source and sink arcs. This completes the
proof.
Q.E.D.
3. \n algebraic study of flows and tensions
Each flow considered ubovc is u vector in Z". Flows could also be considered
'n any ring R, with
@ s, t e R »• j + / e R,
B) Oe/? (zero clement),
C) seR — -seR,
W) s,teR — s.teR
The space Z is not a "vector space" on Z, because Z is not a "field", but
'"module" onZ. and

90 GRAPHS
», te " ~ s + t - {s, + I, jm + /JeZ",
AeZ.seZ » -U ■ (Av, /isJeZ".
Consequently, the set </> of all flows in graph (i constitutes a suhnunlule o
Z"\ i.e., we have:
cp1, cp2 s </> ^ cp1 + cp2 € * ,
Theorem 3. Let G ■ (X, U) he a connected graph. Let II » {X, V) he
spanning tree of G. Denote th' arcs of U — V hy I. 2 k. and denote th
associated c)vles of II hy u\ u8,.. \i". A flow cp h uniquely dejimd from th
values<P\.<Pj, ....«/>* hy
Consider the vector
jp' m (p —
I-
This vector <p' Is a flow, since it is a linear combination of flows. Clearly '
takes only zero values outside of tree //.
Let W c V be the set of arcs i for which <p\ 0. We shall show that W
i.c., each connected component C of the partial graph (X. W) reduces to
single vertex. If not. C is a tree, and from Theorem B, Ch. 3). C necessarily
contains a pendant vertex a. If. for example, the pendant arc is incident into
a. then
which is a contradiction.
Q.E.D.
Corollary. A necessary and sufficient condition that a vector <p he a flow Is
that it is of tin Jornt
ip m jt, n1 + jj u2 + •■• + sk n*,
where st, st sk s Z and u1, \iJ u* are elementary cycles.
Since a cycle is a flow, this linear combination of cycles is a flow. Che
converse follows immediately from Theorem 3. because graph G may be
assumed to be connected (otherwise, each connected component could be
considered separately).
Q.E.D.
In piirtieular. Theorem 3 shows that a cycle u can be obtained by the addi*

FLOW PROBLEMS 91
tion of d" the cycles n' whose "out of tree" arc is used by \i. In this addition,
Cycle is preceded by a + sign if the cycle is in the direction of n. Otherwise,
the cycle is preceded by a - sign. Hence, Theorem 3 reduces the number of
unknowns for the determination of a flow from m to m - n + I.
Theorem 4. A necessary and sufficient condition that a vector <p be a flow
with no negative components is that it is of the form
<p - s, nl + s2 H1 + "• + J» H* ,
yrlwre st, s2 ** e 2, .*„ .*3 sk > 0 and n1. na \i" are circuits.
Clearly, a vector ip of the indicated form is a flow > 0.
Conversely, consider a non-zero flow <p > 0, and let C be the graph
obtained from G by removing all arcs /' with </>, - 0. Graph C contains
no cocircuits. and therefore by the arc colouring Lemma.it contains at least
one circuit u1. Let.?, > 0 be the smallest flow in an arc of u1. The vector
•P1 ■«(» - *iH'.
is a flow > 0 that has more zero components than ip. If <p' is not a zero
vector, repeat this process, etc. ... Finully, a /.cro flow of the form
cpk - <p - j, n1 - sj n2 - - - s» n* - 0.
is obtained with ;,, ;a sh Z and sit s.2 sk > 0.
Q.E.D.
A tinsion (or potential difference) is defined to be a vector 0 - @,. tf,
0ra) 6 Z" such that, for each elementary cycle /<.
I Of I 0i.
This eqiiiility can be restated by saying that the scalar product < ji. 0 -
2 /<■ 0, is zero.
Let 0 denote the set of all tensions. Note that 0 is a submodule of Zm. i.e.,
O'.O'e© 01 + OJ
*- .@6 0.
I'licorcin 5. A vector 0 ■ (f^. fti ft,,,) Is a tension if, and only if. there
exists a function t(x) defined on the vertex set X with values in Z suck that,
for each arc i - (w. />). 0, - /(/>) - /(«).
The function t{\) is called a potential attached to tension 0.
'• IfH is a vector defined by a function t(.v), then consider the cycle

92 ORAPHS
that successively encounters the vertices a, b, c, ..., z. Then, we may write
tii, 0lt - i(b) - t(a)
M, Oi, - /(c) - t(b)
Adding the above equations yields
I 0t- I 0,-0.
iip* it*-
2. It is easy to calculate successively the coefficients i(x) for a given tension
0, by the following rules:
Rule I. Take any vertex .v0, label .v0 and set
'(*o) - 0
Rule 2. If .v is labelled and y is unlabellcd, and if /' — (x, y) is an arc, put
Ky) - /(*) + 0,.
If /-(>'. .v) is an arc, put
In this way, all the vertices ofn connected graph will be labelled. (If the graph
is not connected, each connected component can be treated separately.)
Each coefficient is uniquely defined by this process. Otherwise, there would
exist two chains n1 and \ia from .v0 to .v such that
and consequently.
Since \il ■ na is a flow, it is a linear combination of elementary cycles by
virtue of Theorem 3. Therefore, there exists <in elementary cycle u such that
which contradicts the definition of a tension.
Q.E.D.
Consequence. This theorem clearly shows that a cocycle w(/f) is a tension,
since we may let
0 if xeA
1 if xiA.
Hence, for an arc / - (a, b).

FLOW PROBLEMS 93
[ + 1 if I e o>+(A) \
t(b) - t(a) - - 1 if leo)-(A) - o>t(A).
I 0 if /#w(/l) J
Theorem 6. Let G ■ (X, V) be a connected graph. Let H ■ (A', V) be a
spanning tree of C with arcs I, 2 /, let o)\ wa to' be the cocycles asso-
associated w'lh H. A tension 0 - (fl,, fl2 6m) is uniquely defined from the wlues
0,,fl3 0,by:
e - o, oI + o, to2 + ••• + o, w'.
The vector
0' - 0 - 0, oI - fl, to2 - ••• - 0,«'
is a tension that has zero value on each arc of the tree //. I f 0' corresponds to a
potential /'(*), then
/'(*,) - /'(*,) - - - /'(*„).
Consequently, 0" - 0.
Q.E.D.
Corollary, A necessary and sufficient condition that a vector Obea tension is
that it is of the form
0 ■ sx (iI + s2 (iJ + "• + sk (o*,
where st, sa sk eZ, and co1. coB w* are elementary cocycles.
Cle-irly, each linear combination of elementary cocycles is a tension, and
conversely, each tension is a linearly combination of elementary cocycles,
from Theorem 6.
Q.H.D.
Theorem 6 shows that n tension
O-@..0j 0m).
°an be determined from /l(C) ■ n - I unknowns.
Hjcorcm 7. A necessary and sufficient condition that 0 he a tension > 0 is
that
0 ■ j, (iI + st to2 + — + sk @*,
'here sx, ja,.... gk e Z, su sa,.... sk > 0 om/ w1. wa, .... co" arc elementary
focfrcuits.
Clearly, any linear combination of cocircuits with all coefficients > 0 is a
tcnsion > ().

94 ORAPHS
Conversely, consider a tension 0 0. 0 > 0. We shall show that there
exists an elementary cocircuit co1 and an s, > 0 such that the vector
0 - s, oI
has more zero components than the vector 0.
Let / - I be an arc with
0, - min{0,/0, * 0. 1 */*/«}.
Put 0, - *, > 0. Colour black ill arcs /such that (I, > 0. Colour red all arcs/
such that 0, - 0. There cannot exist a red and black elementary cycle with
all black urcs in the same direction that contains are I. Thus, by virtue of th
Arc Colouring Lemma, arc I is contained in a black elementary cocyclc co1
with all urcs in the same direction. Thus, co1 is a cocircuit, and the vector
0 - s, oI
has more zero components than the vector 0. This vector is also a tension
If 0 - .?, co1 0, this reduction pruccss can be repeated until a tension
0 - s, (iI - s2 to2 - •• - sk w* - 0.
is obtained.
Q.E.D.
Theorem 8. A rector <p e 2" is a flow if, ami only if, it is orthogonal to each
rector of 6. A rector 0 c 2" is a tension if, and only if, it is orthogonal to each
vector of <f>.
(Hence, 6 and <t> arc two orthogonal subinoclulcs of Zn.)
I. We sh;ill show that if «p <t> and (le (•), then 0 and <p arc orthogonal,
£ *,0
i-i
For each elementary cycle >i.
From Theorem 3, ip is of the form
«P -
Thus.
<<P-0>- I ** < M*. 0 > - 0 .

ILUW PUnlll EMS 9$
2. Let «p be a vector such that <<p. 0> « 0 for all 0 0. Vector <p is a
flow, because if we take 0 w{.v) for some vertex *, then
I Vt- [ Vt - <(■>(*), cp> -0.
liu'M Iiu'(j)
3. Let 0 be a vector such that <(p. 0> - 0 for all (pe<l>. Vector 0 is a
tension, because for every elementary cycle u. we have
<u,0>-0
Q.E.D.
Additional algebraic results for flows and tensions appear in Bcrgc and
(Jliouila-I louri [1962]. uiul Slepian A968).
4. I he maximum tension problem
Consider a graph G whose arcs arc denoted by 1.2 m. Let A,. A-a
km, ii.it /„ e 2, such that
-oo**(</, <+co.
Scvcr.il problems, similar to the above problems can be stated for tensions.
Hie compatible tension problem. For graph G, Jhul a tension 0 such thai
k,<0,<l, (/-I, 2 /«).
The maximum tension problem. For graph G. find a tension 0 such that
A) kt « 0, «/, (/- 1.2 /»).
B) tf, is maximum.
E I. The shortest path problem (Ch. 4, § 2) is a special case of the
maximum tension problem. Let a and /; be two vertices of a graph G". Let
l(x,y) denote the length of arc (x,y). Add to G an arc I - (/>. a) and let /(a)
denote the length /(>i[«. x]) of a shortest elementary path fi[a, x] from a to x.
For each arc (x, y) of G, let
0(x,y)-t(y)-t(x).
Consequently,
-oo < 0{x,y) <l(x,y).
We wish to maximize
0(b,o)-/(fl) -/(<>)- -Kb),
s'ncc /(ft) must represent the length of the shortest path from a to b.

96 urapiis
Example 2. Sequencing problems: The construction of a factory require
the performance of various distinct tasks designated by xuxa, ...,xn. Bcfo
starting a task x,, it is usually required that another task xt be sufficiently
underway. Let k,, denote the amount of time by which the start of task < must
precede the start of task /.
Furthermore, a task x, cannot begin before a fixed time k\ and has a known
duration k"t. Given these constraints, when should each task begin so that
the construction project finishes as soon as possible?
Consider a graph G whose vertices correspond to the tasks. An arc (x,, x,
is present if task x, can start only after task .v, is sufficiently under way. Add
source a and an arc (a, x) for each task .v that cannot begin before a specified
time. Finally, add a sink /; and an arc (.v. b) for each task x.
Denote hy r(.v) the starting time of task x. Let
t(a) - 0
We wish to minimize /(/>), i.e.. to maximize the tension
0(b,a)»t(a)-t(b)
in the "return arc" (/>, a). The constraints arc
k\ < f(*i) -0 -fl(o,x,) < + oo if (a,x,)eU
k,j < /(x;) - f(x,) - 0(x,,x;) < + oo if (x,, Xj) € U
k? < t(b) - t(x() m 0(x,,b) < + oo if (x,, b) £ U.
The solution of sequencing problems as potential problems has been
notably developed by B. Roy [1965].
Lemma. A necessary condition for the existence of a compatible tension is
that for each cycle fi,
I /i - I *, > 0 .
In fact, if such a tension 0 exists, then for each cycle p
0 - < n. 0 > - < ,i+. 0 > - < M~, 0 > > £ fc, - £ /,.
Q.E.D.
The sufficiency of this condition will be demonstrated later.
Compatible tension algorithm (J. C. Herz [1967])
Let C be a graph with the numbers A:, < /, associated with each arc /. For
a given tension 0. compatible or not, define the distance of 0, from the interval

FLOW PROB1.IM.S 97
f-0 if 0,e[fc,,/(],
</,@,) - fc, - 0, if 0, < k,,
I - 0, - I, if 0, > /,.
We shall successively reduce the quantity
If r/@) - 0. the tension 0 is compatible, and the procedure stops. Other-
Otherwise, there exists an arc i with </,(#,) > 0; let
and suppose that
rf,@,)-A, -0, , 0, <fc|.
To construct a tension 0' such that </(G') < </F). successively label the
vertices of 6' in the following way:
Rw f I. Label vertex a. the terminal endpoint of arc I. with the ir.dex + I.
Rule 2. If .v is Libelled and y is unlabellcd. label y with the index +/ if
ix.y) - /and if 0, < it,.
Rule 3. If a- is labelled and y is unlabcllcd. label y with the irdcx -/ if
(>■. v) - / and if 0, > /,.
If vertex A, the initiul endpoint of arc I,cannot be labelled by this procedure,
then the tension 0 can be improved. In fact, the set A of labelled vertices
satisfies
aeA. biA.
The tension 0' - 0 u>M) satisfies #;-#, + !, andO' compatible, because
'ew'H) implies 0, > it, and l)\ > *,. Similarly, lew (A) implies », < /,.
andO; a /,.
If this liibclling procedure labels vertex b. then there exists a chain
v - [a,alta2, ...,*],
in which each vertex has been labelled from the preceding one. Thus
lev* ^ 0, <*,,
i e v" ~ 0, > /,.
f ■ v[fl, /,] + [bt a] is a CyC|c. since 0, < it,, we have

98 GRAPHS
o-<o.M>- £ o,- £ e,< £ k, - £ i,.
I a It' lt*~ Ian' len~
From the preceding lemma, it follows (hat no compatible tension exists.
The algorithm yields the following result:
Compatible tension theorem (Ghouila-llouri [I960]). Ciuen a graph (! an
numbers k, and /, where — oo < k, < /, < + oc for i — 1.2 in. a nece
sary ami sufficient condition that th're exists a nnsion 0 ■ (W,. 0 9
with k, < fl, < /, for all i, is that for each cycle fi,
£ /, - £ fc,>0.
t*nf i*n~
rhc necessary part or (he theorem follows from the preceding lemma.
The condition is sufficient because, then, each tension 0 with </(()) > Ocan
be improved using the preceding algorithm until we obtain tension 0' with
</((>') - 0.
Q.E.D.
Corollary 1 (Roy. fl%2J). There exists a tension 0 with 0, > kjor all i. if,
and only If./or each circuit >».
The proof is achieved by taking /,■ +co for all i.
Corollary 2. Tin re exists a tension 0 such that 0, < /, for all I. if, and only if,
for each circuit yi.
I I, >0.
i*h
Corollary 3. A vector
V -O'l.^a.-.^J
is called a subtension if there exists a tension 0 such that yt < ff, for all i. A
necessary and sufficient condition that a rector y to a subtensim is that
Furthermore, a suhunsion y and a flow ip > 0 satisfy < «p. y - 0. if. and
only if, y is a tension on the partial graph generated by the arcs I with <pt > 0.
I. If y satisfies the above inequality, then by taking for «p si circuit u, we
obtain
< m. y > - I y, < o.

FLOW PKOniCMS 99
Corollary I, y is therefore a subtcnsion.
i Conversely, let y be a subtcnsion. If «p e <J\ «p > 0. then
< «p.y > - L <P(}'i < £ </>ify ■ o.
<-i <-i
|-urthcrmorc. equality holds if. and only if.
</>, > 0 ^ y, — W,.
Clearly, if \ <p. y / "• 0. then y is a tension on the partial graph generated by
the arcs / such that </>, > 0.
I'inally, if y is a tension on the partial graph generated by the arcs / with
tfl( > 0, then
I >'i <f>i - 0.
nnd, consequently,
«ii > 0 Pi» 0
Q.H.D.
Muximum tension algorithm
We shall construct a tension that maximizes the value 0t of the tension in
are 1 - [b, a).
Siiirting with any compatible tension 0. achieves the labelling procedure for
the compatible tension Problem. If vertex-/> cannot be labelled, the set A of
labelled vertices satisfies a A, b$ A. and the tension 0" - 0 - co(/f) satis-
satisfies f)[ m 0t + I. Thus, the vulue of the tension in arc 1 can be improved.
If vertex ft can be labelled, we shall show that the value 0, of the tension 0
in arc I is maximum.
lorcuch compatible tension 0 and for each cycle }i that uses arc I along its
direction, we have
0-
or
lull" I a/i'
1*1
algorithm labels vertex />, then there exists ;i chiiin v[«. b) such that
iev* *• 0, -A-,,
/6V~ ^ flj-/j-

100 URAPIIS
Hence, for the cycle yr » v[a, b] + [b, a], we have
0, - I 0,- I *,- I /,- I *,.
It ltm l«n* '•*" lap*
1*1 1*1
From inequality (I), we sec that 0\ has reached its maximum value.
Theorem 9. f'vr a graph C with arcs numbers k, ami /„ the maximum value t
compatible tension in arc I is
{ ■■ min / £ /, - £ /c,\.
1 «J«# \ 1*1 /
! < min / X '' ~ Z k<
i Tu» V *#i
max 0.
If 0 is a compatible tension with
0.
it can be improved by using the above algorithm. 1 lie theorem follows.
Q.E.D.
EXERCISES
1. Let
S" ~(Ti.T2....,Tn)
be a partition or a finite set X, and let
5"-(Si,Sj Sn)
be a family of subsets of X. Show that If every union of & of the T, contains at most k
the S, (for km |,2 n), then lhere exist indices It, t» tR such that
TtpnS, a for /» — 1,2 m.
Hint: This result is easily shown by constructing the appropriate transportation network.
2. Let
£"-G-..r, r«>
be a partition of a finite set X, and let
^-(S,.Sj 5n)
be a family of subsets, and let o, ca cn be positive integers. Show how to constr
a set of representatives
A - {fl|,flj,...,fln }.
such that fl! s S1, at e Sa,.... etc, and such that
Mn7/l<pj pour y-1,2 «.

Fl OW PROB1 EMS 101
Hy reducing this problem to a flow problem, show thai such a set of representatives
cxi*» if
I U S<n U T, I > | /1 + £ t T, | -£ e,
l I
for all / c I '• 2 "'' ond a" -f c { I. 2 n}.
J. Let
^"-G-|,7, 71,)
be a partition or a finite set X, and let
-(Si.Si SB)
be u family of subsets, and let h,, b3 bn be positive integers. Show, how to construct
it set of representatives
A - {ai.ai an },
with <u Si. d £> and such that
\AnTi\*b, for y- 1.2 «.
Show that such a set of representatives exists If
\\JS,n\jT, I > UI — m + XO
for all / c ( |. 2 m} and for all J <= { 1, 2,.... n}.
4. U\e ihe Compatible Flow Theorem to prove the following:
Let R be a transportation network with source a and sink b. Associated with each
source arc Ic u'Ui) are two numbers b, and c, such that 0 < ft, < «■,. Associated with
each sink arc / e w "(/>) are two numbers*,' and e', such that 0 < /»; < c',. A necessary and
sufRcient condition that there exists a flow <p satisfying
■> that both
A) There
B)
There
of the
exists
exists
following
a flow (p1
a flow <pa
< % < r;
conditions
with
with
for
for
hold:
for
for
for
for
y 6
/ s w
lca>'{a)

CHAPTER 6
Degrees and Dcmi-Dcgrces
1. Existence of a />-grnph with given di'inl-dcgrvvs
For a graph U - (X, I'), the outer deml-degree </<•(*) of a vertex
defined to be the number of arcs having .v as their initial end point, i.e.
de(x)- I mj(x.j-).
Similarly, the inner demi-degree d{j(x) of a vertex .v to be the number o
arcs having .v as their terminal endpoint, i.e.
i?x
Finally, we define the degree do(x) of a vertex .v is defined to be the intege
Thus, a loop at vertex v increases the degree of vertex .v by 2.
A p-graph is u graph with wj(.v. y) < p for all vertices x, y. (Jivcn integers
r\. ra, ,.,,rH, ,v,,ja .Tn. we may a.sk if there exists u /j-gnipli A with
vertices .Vn Jfa,.... .vB such that
rffiU*) - i-* (*-l.2 n)
dcM m Sk (k m |,2 n).
In this case, the pairs (rk, jk.) are said to constitute the dvmi-degrees of a
graph.
Theorem I. Let (/■,, *,), (ra, sa)..... (rm, xn) be pairs of integers with
•v, > s2 > - > *B.
The pairs (rk, sk) constitute the demi-degrees of a p-graph if, and only if.
n *
A) Tmin{r,,pk,}>Ysj (fc-|,2 n - 1)
|S| /-I
102

DEGREES AND DFMI-DEGREES 103
B) I."
Construct a transportation network R with vertices .v,, x xn, jc,. .va,
£,, and with a source a and n sink b. Join vertices x, and .v, by an arc with
Opacity r(.v,, if,) - p. Join vertices a and .v, by an arc with capacity c(a, x,) -
r Join vertices 5, and /) by an arc with capacity r(.v/( /)) - s,.
Any flow that saluratcs the source and sink arcs of R defines a ^-graph
(Jk'. V) having dcini-dcgrccs (rk,*fc). Conversely, each /vgruph (X, V) with
dcini-dcgrccs (rk,sk) defines u flow in R that saturates the source and sink
arcs, l-'rom Theorem B, Ch. 5). a necessary und sufficient condition for the
existence of such a flow is that condition B) hold und that, for each
set B m {-?i|i -fia -Pi,K the total flow that can enter B is greater than the
total demand at B, i.e.,
F(x,,.x(| xlk) > (i{xh,xit x,,) A < it < ij < - < ik < n).
'1 his is equivalent to
n
(I1) £ min { r,, pk } > *,, + s,, + •• + s,h A < (, < i2 < - < i» < m) .
i-1
Clearly (D A) and. since the i, arc indexed in decreasing order,
A) (I1).
Q.E.D.
l;or p m I. these necessary and sufficient conditions can be restated in a
simpler form. Let
u //-tuple of integers. Corresponding to this //-tuple, associate the sequence
(r*. '"*■...) where /•? denotes the number of r, that tire greater than or equal
to the integer k. Hence.
To visuali/c the numbers r*. we can construct a diagram, culled the
Ferrers diagram, as in Mg. (>. 1.
Match the first r, squares in the /-th column in the positive quadrant. '1 hen
't is easily seen that /•; equals the number of hatched squares in thc/-th row
ln the positive quadrant. By counting, in two different ways, the number of
•latched squares, we obtain:
sequences (/•,) and (rf) arc called conjugates. Using these definitions,
several corollaries follow from Theorem I.

104
GRAPHS
pk
0 12 3
i-1
Fl«. 6.1. Ferrers diagram for ihe sequence (/•,) - E, 4, 2, 2, I, 1). Mere,
(;•,•)-F.4, 2, 2. I)
Corollary 1. Given n pairs (r,,s,) of integers such that
$i !* st ^ '" ^ *«»
there exists a p-graph with r/J(x,) - r,, </j(.v,) - j,,/or / - 1.2 « //".
i-i J"i J
By counting in two different ways the number of hutched squares in the
first pk rows of the associated Ferrers diagram for (rt ,r rn) we find that
n pk
£ min { r,, pk } - Y r,* .
Q.E.D.
Corollary 2 (Ryser [1957]; Gale [1957]). Consider n pairs (r,. j,) of integers
such that
7Viprtf f.v/.ff.v a 1 -j?ra/>A 0" with r/J (x,) - r, and d ,"(.v,) - *f if, and only if,
/■;> £ J; (fc™ 1.2 n- 1)
I
i-i
lit r^
f"* - L 5i •

DEOKFE3 AND DEMI-DEOKKES 105
This follows immediately from Corollary 1.
Corollary 3. Given a sequence {rltr rn), there exists a l-graph G with
.(Xl) m r, and d£(x,) - I for all t If, and only if.
This condition is necessary because if such a graph G exists, then
n n
Z r> ■ Z <*</(**) -" •
l-l l»l
Conversely, assume that the condition holds; by Theorem 1, we have only
to show that, for all k < «,
n
Z min {k,r,} > k .
inhere exists an r, > A-, the above inequality is satisfied. Otherwise, each r, is
less than k, and consequently.
n n
Z min { k, r, } - Y r, ■ n > k .
1*1 i-1
Thus there exists a graph G with the desired properties.
Q.E.D.
We shall now present necessary and sufficient conditions for the pairs
('i. *"i). (/"a. ra) (/•„, rn) to constitute the demi-degrces of a symmetric p-
graph. i.e. a /'-graph G with
»»<;(x. y) - >Hc(y.x) (x,yeX).
Theorem 2. Let Go he a symmetric graph such that each odd cycle contains a
vertex with one or more loops, and let rx, ra rn be positive integers. If Go
'wt a partial graph H with djj (*,)•• dJUx,) - r, for each rert >x x,, then Go has
a symmetric partial graph G with
r,
for all i.
We shall assume that the theorem is true for any graph Go of order < n, and
wc shall show that the theorem is true for a graph Go of order ii. Given a
8raph //. we slull construct from // the symmetric graph G.
'• Suppose first that there exists a vertex .v0 such that

106 GRAPHS
Consider the subgraph Go of order n - I obtained from G'o by rcmovin
vertex x0. Graph Go has a partial graph TJ with given dcmi-degrccs ,
ft - »»i}(x0. *•)• which can be transformed to a symmetric graph G wit,
dcmi-degrces r, (by the induction hypothesis). This yields graph G.
2. We mny now suppose that for each vertex x of Go. there exists a vertex
adjacent to x with
mfa.y) > mniy.x).
Since d}j(x) - rf,7(.v), there exists a vertex 2 adjacent to x with
Clearly ,v, y and r arc distinct vertices,
3. Choose any vertex .\|, and let .va be any vertex such that
nt^x,, xj) > »M,)(xltx,).
Let x3 be any vertex such that
In this way, n sequence .V|..va. .V3.... is defined and, since the graph \
finite, an elementary cycle
/4 ■ [X^ , Xj,+ |, ..., Xj,+jt_|, Xp+|( ™ Xj,]
will be formed.
4. If cycle n is even, transform // to //' by removing an arc between each
of the pairs:
and adding an arc between each of the pairs:
This docs not alter the dcmi-degrces. and produces a graph //' with
A) XI »'«•(*. y) - '««•(>'. x) I < X I m»(*. >') - "»(^. x) I •
x,y«X x.yX
x+y x*y
5. If cycle n is odd, and if // has a loop ut one of its vertices, say xv, the
transform // into //' by removing an arc between each pair
and adding an arc between each pair
(Xj,+ |, Xj,), (Xp>j, Xj, + 1). ..
This does not alter the dcmi-degrces. and the graph //' .satisfies (I).

DEGREES AND DI'MI-DFGREEJ 107
6 Finally, if cycle >< is odd and if // has no loops attached to ft, then
there exists a vertex of ft. say xp, that is incident with a loop in G"o. Transform
// into //' by adding an arc between each pair
and removing an nrc between each pair
Again, this docs not niter the demi-degrecs, and graph //' satisfies (I).
After repeating this procedure a finite number of times, a symmetric graph
C is obtained.
Q.E.D.
Corollary. Given Integers r, > ra >'••> rK, a necessary and sufficient con-
condition that there exists a symmetric p-graph with
for ail I, is that
i* i
Z rt > Z ri (fc-1.2 m-I).
1-1 y-i
1 he result follows by applying Theorem 2 to the />-gruph (!0 with vertices
X\, .va>.... ,vn and p arcs going from .v, to .v, for all / and all /. and then, invok-
invoking Corollary 1 to Theorem I.
The following consequence of Theorem I is used to characterize tourna-
tournaments.
llu'orcm 3. (Landau [1953]; Moon [1963]). Tixere exists a complete anti-
antisymmetric \-graph with outer dcml-degrccs
if, and only if.
(fc-2,3 h - I)
f \ denotes the binomial coefficient ., P '—rr •
ihe condition is netvxsary because, in a complete anti-symmetric 1-graph
c ni|mbcr of edges joining the vertices of the set {xt. xa xk) is less than
Or equal to the number of arcs leaving the vertices of the set.

108 GRAPHS
The condition is sufficient. To prove sufficiency, let
s,-(n-\)-r, (/-I, 2 n).
Thus,
*l > *2 > *J > ■" > *. •
1. First we shall show that the conditions of Theorem 1 are satisfied for
p m 1. Note that
£ r, - n(n - 1) - ^^ - - £
Furthermore, by letting ik denote the number of r, that are < k.
s, - n(n - 1) - £ r, - n(n - 1) - -^r—^ - -~—- - £ r,
£ min { k, r, } - £ r, + k(n - tk) > ('*)+ k(n - lk) .
i-i i-1 \*/
Note that, for any integers A; and /.
2 [^) + Q - «' - ')] - t(t - 1) + k(k - 1) - 2 fc» + 2 fc
- (fc - 02 + (fc - 0 > 0 .
Thus,
f min { k, r, }> ('*) + k(n - tk) -
(-i \*/
> k(n - 1) - (^)
>««-!)- f r, - £s, ,
1-1 1-1
and the conditions of Theorem 1 are satisfied.
2. From Part 1. there exists a I-graph G ■ (X, V) such that
dc(x,) - r,. dc(x,) - s, (i - 1, 2 n) .
If this l-griph G has neither loops nor two oppositely directed arcs joining
the same pair of vertices (call such nrcs multiple edges), then G is a complet*
anti-symmetric 1-graph because the number of arcs is ^[*_, r, - ( ). Inthi*
case, the theorem has been verified.

DEOR U AND DEMI-DEGREES 109
Otherwise, we shall alter G and decrease the number or loops and multiple
joes without changing its demi-degrecs. Suppose that at vertex x, there are
gjx) loops and q(x,) multiple edges. The number of vertices that are not
adjacent to x, is
Colour red each multiple edge and loop, and insert a green edge between
each pair of non-adjacent vertices; then the number of red edges incident to
x equals the number of green edges incident to x. Suppose we travel through
the coloured edges of the graph without using two edges of the same colour in
succession and without using the same edge twice. Then, after arrival at a
vertex x, we can always leave x except perhaps if x is the initial vertex of
the tour.
In other words, if there exists red edges, then there exists a cycle
with alternately red and green edges such that
v.
V.
Thus G may be altered by removing arcs OWi). O'«. ya\ etc... and by
adding arcs (ya,y3). (>W<>) O'a*.>'i). without changing any dcmi-
degree. This process decreases the number of red edges. It can be repeated
until no more red edges arc present.
Q.E.D.
2- Existence of a /7-graph without loops and with given dcmi-dcgrccs
Consider the following problem:
Given a graph Go - (X, V), construct a partial graph If with given demi-
bgrees </,}(*) and </,7(x).
For A c x and B c X, let m$0 (A, B) denote the number of arcs in Go
*hosc initial endpoint is in A and whose terminal endpoint is in B. If an
integer r, is associated with each x, e X then, for each A <= X, let
K/0- I r, .
X,§A
Iheorcm 4. For a graph Go with vertices xitx xn, and integers r,, s,,
^'■1,2 n, a necessary and sufficient condition that Go have a partial
II with

110 ORAPttS
<*;;(*i) - r,, ■ dj,(x,) - s, (/ - 1, 2 n)
is that
A) t min{r,,<,(x,./0}>5M) (A <= X)
B) I I
Consider n bipartite transportation network R - (A'. X. V) with vertex;
sets X m {Xi, xa,..., xn} and ^ - {XltXa, ..., X,}, with a source a and
sink A, and with arcs
(*,, X,) with capacity m$0 (x,, xt)
(a, x,) (For I < / < n) with capacity r,
(AV, A) (for I <./<//) with capacity J,.
If condition B) is satisfied, the desired partial graph // exists if. and oly
if, network R has a maximum flow that saturates the sink arcs; from Theorem
B, Ch. S), this is equivalent to
(I1) F(A)>d(A) (AcX),
where F(A~) is the maximum amount of flow that can enter J, i.e.
n
and <i(J) is the total demand of set A, i.e.
Clearly, condition (I') is equivalent to condition (I). O E D
Corollary 1. Given pairs of integers (r,, $,), (r3. jj). ..., (rn, $„). o necessar
and sufficient condition for the pairs to constitute the demi-degrees oj a p-gropk
H without bops, is that
(I)
imm[r,.p\A-ix,)\}>s(A) {A c X)
We apply Theorem 4 to a complete symmetric />-gr;iph Co without loopfc
Clearly,
m£jLx,.A) - p\A - {x, )\ .
The result follows. Q.U.D.

DEGRFM AND DFMI-DCUIE 8
III
Tlicsc conditions simplify considerably when
r2
s2
Consider n sequence (r,, rt /•„) of positive integers such that
I et r\ denote the number of indices / such th.it / < k and r, > A - 1 plus
the number of indices / such that i > k and r, > k. The sequence (/\) is called
the corrected conjugate of sequence (r,). The numbers Fk can be visualized on
the corrected Ferrers diagram (Fig. 6.2). formed by dividing the positive
qu.Klr.int into three parts: hatched, dotted or empty.
All squares on the principiil diagonal arc dotted. In the i-th column, the
first 'i squares not on the diagonal arc hatched. All other squares arc empty
k
■ • * *
1
^ t
■ '.'i'i
4
^\
7 t
1
k
Mr. 6.2. Corrected Ferrers diagram Tor the sequence G, 5, 5. 5, 5. i. 2, I). The
corrected conjugate sequence Is G, 6, 5.4,4, 5.1, I)
Clearly, the number of hatched squares in the *-th row of the diagram
equals Fk. and therefore

112 GRAPHS
Corollary 2. Let (r,. s,) be pairs of integers with
Let (rt) be the corrected conjugate of{r,). There exists a 1 -graph II without loops
with d,'t(x,) m r, anddf^x,) m s,/or all /. anil only if.
A)
B)
1.
If such
*
,?,'■
a 1-graph
n
V min {
k
n
exists.
r,.\A
s,
s,
then
-{
(k
x,}l
-
}
1.
>
2,...
s(A)
.»- 1)
(Ac
■
X).
By taking /4 - {.v,,.va xk}. condition A) follows. Condition B) ob-
obviously holds.
2. Conversely, suppose that conditions (I) and B) arc satisfied. For any
set A of cardinality k,
n
£ min { r,. M - { x, } | } -
k *
- £ min { r,, fc - I } + £ min { r,, fc } > £ r, > £ s, > s(-4).
F-'roni Corollary 1, there exists a graph // without loops with </,*(*,) 'i
and ttfiix,) ■ j, for all /.
Q.F..D.
We shall now consider necessary and sufficient conditions for the p.»ir»
(ri. fi). ('"a. ''a) ('■». ''n) to constitute the dcmi-dcgrccs of a symmetric f
graph without loops, i.e. a ^-graph with:
'«o(x, y) - m£(y, x) (x. >■ e X).
First, we shall prove the following very general result:
Theorem5(Fulkcrson, Hoffman. McAndrow [1965]). LetG0 - (A\ U)be
symmetric graph of order it without loops, such that any two vertex-disjoini eft"
mentary cycles >f odd length are joined by an edge, let r,, ra,..., rn be integers
whose sum is even.
If Go has a partial graph il wflhdjUx,) — dij(xt) ■■ r, for all /. then Gohas
symmetric partial graph G with

ohikms and 01 mi-di.orfts 113
dc(x,) - </J(x,) - r, (/ - 1. 2 n).
I. Let
/i(x, v) - w/,(x, y) + m,)(y, x).
(I) /i(*.r)
>0 < 2
C) I /i(x,.>) - 2r,.
If the numbers/(.v.^) .»rc all even, then the graph G defined by
'n!i(x,y) - -• f(x.y)
is the required graph.
Otherwise./,(.v. >') determines n non-empty set A', c a{X) defined by
Ex - {[x. y] I [x, y] s 2m, f,(x, y) m 1 (mod. 2)}.
Let //] - (X, Ei) denote the simple graph having Et as its edge set. Note
th.it in //i all degrees arc even, because
<*«.(*»>■ I /i(**.>->-2r4»0 (mod. 2).
2. We shall now successively construct a sequence /j ,/3 /„ of functions
of two variables that satisfy (I). B). and C) and define (as above) sets of
edges
£, 3 Ei 3 - 3 E,
a»d simple graphs //|,//a //,. When we encounter a set £, - 0
('•c, a function/,(.x-.y) with even values that satisfy conditions A), B), and
W)h we shall obtain the required graph C. defined by
>) //)
3. If the simple graph /Y, has an edge, then it has a cycle (because all its
degrees are even). Let n denote such a cycle.
"" cycle fi is even, define a function fa(x,y), that differs from /,(*. y) only
°n tlic edges of n, by adding alternately + I and - I to/i(x.>) while tra-
^fsing cycle p.
Since cycle n is cvcn./iU.y) also satisfies conditions (I). B) and C). and
"••ermines a new set of edges £a " ^"i - /'• The simple graph //a - {X, Ea)

114 GRAPHS
defined in this way has only even Jcgrccs. If //] contains an even cycle
we can similarly define //j - (.V. £3). etc. Repeat this process until a simp|
graph II,, - (X, Er) without any even cycles has been found.
4. If the graph llr has an edge, it contains a cycle n (because all its degr
arc even). This cycle n is necessarily elementary (because otherwise the cd ,
of/4 would contain an even cycle which contradicts the definition of //,, n
odd. Let
be this elementary odd cycle of //„. Then
y /.(«f, «,., |) s 1 (mod. 2)
i-1
and
I
]L fpixi, Xj) ■ - y j^jpixi, x^ ■■
- £ r, ■ 0 (mod. 2) .
This shows that/r has at least one odd value on an edge not in fi, and thai
//,, contains an edge not in fi. Thus //,, has another odd elementary eye
v fi, and r has no common vertices with /< (since, otherwise, they would
form an even cycle).
From the assumptions on Go, these two cycles arc joined by nn arc of 0
For example, let
Edge [fli, /)|]. which is not in IIft satisfies
//<!,.*,)  (mod. 2).
M " 0. the functionyj,,, is obtained by letting/r.,(«,,
and by adding alternatively - I and + I to the edges [ax. aj], ...
and nlso to the edges [/>,. /)a] (/>a/f i« *il-
lf/.(fli. /'i) 0. let/,., ,(fl,. />,) -/,(<?!, />i) - 2 nnd add alternately
and - I to the edges (fli.fla) (tfa*+i.fli) antl t0 ttlc edges [/>!, /)a).
»ai + i.*i]-
Function /p4, also satisfies conditions (I). B) nnd C). and it yield

DKiHI'FS AND DFM1-DF.CRFES IIS
thc simple graph //„,, - (X. /'p» i). where £„ +1 - Ep - (u u v). Since E is
fnitc we obl.iiii in a finite number of steps a set E, - 0.
Q.F.D.
Corollary. Given the integers p > I, r, > r2 > - > rB > 1 .v//r/;
s>, ft <''<'«. « necessary and suflick it! condition for the existence of a sym-
symmetric p-graph G without loops whose vertices satisfy d,*(x,) - */G(.v,) ■ r,,
is dull
n
£ min { r-|. p H - { x, } | } > £ /•, 04 «= X) .
J ■ 1 *r f A
The proof follows by applying Theorem 5 to a graph C70 with vertices
Xl, x3 *■ and with p ares from .v, to .v, for each pair (a,, a1,) with xt x,.
}, ICxislcncc of a simple graph with given degrees
This section describes necessary and sufficient conditions for a sequence of
integers </, > d2 >■•■> dn to constitute the degrees of a simple graph.
Theorem 6 (Erdos. Gallji [I960]). Let «/, > d3 >•••> </n /»c a decreasing
sequence of'n integers with ^ </, wph. «»«/ /<7 (<7,) denoli its corrected conjugate
sequence. The following conditions are equivalent:
A) There exists a simple graph G whose vertices x, satisfy do{x,) — </,:
B) I tit* Z't, (ft- 1.2 ii):
C) £ dt < fe(Jk — 1) + ^ mm { ft-'';) (fc ■■ 1, 2 n).
A) B) Condition A) implies the existence of a 1-graph 0'* without loops
such that (/(/.(.r,) - d&(x,) - </„ for all /: this implies B). by
Corollary 2, Theorem 4.
'2) (I) Condition B) implies the existence of a 1-graph // without loops
such that d,*,(x,) - dfi(x,) - d,, from Corollary 2. Theorem 4.
Since ^ d, is even, this i in plies from Theorem 5 the existence of a
symmetric I-graph G* without loops such that
dff.(x,) — </J.(.v,) — d,.
") •• 0) Consider the corrected Ferrers diagram for the sequence (</,), and
denote by ak the number of empty squares in [0, A] x [0, A);
condition C) is equivalent to
C')

ttfi OH API IS
Since ak > 0, condition B) implies condition C'), which impli
condition C).
C) => B) Suppose condition C') is satisfied and there exists an intcgc
with
I </, > £ I.
j-i i-i
We shall show th.it this results in a contradiction.
Clearly At > 1, Since only one square in [0, I] x [0. I] is dotted,
we have «, - 0 and. from condition C'), dx < 3X + «, <?,
Let q be the largest integer such that dq > k - 1, Then q .
(since ««„ > 0). Hence
£ </, > I«7,-ii(ft- D+ I rf, + I rf,.
I*-1 1» I J-II+1 l"tll
Thus
(i) I ''(></(fc - D+ I tii-
i>i i-jk> i
On the other hand, from condition C). we have
, < q{q - I) + £ inin { d,, </} <
< </(</ - I) + £ min { rf|t </ } + £ </,
i - f 11 j • a »i
<</(</ - 1) + (fc - q)q + £ </,.
l-M I
Hence
(ii) I rf|<«(k- D+ I rfi.
i-1 i»ii» t
Comparing (i) <intl (ii). we obtain the desired contradiction.
Q.E.D.
We shall now describe conditions for a sequence (</,) to constitute ••
degrees of various types of simple gruph.
Theorem 7. Consider two sequences /^ > ra > ••■ > rp ami s, > Ja
... > s,. w///i /) < </. 77«'rr f.v/.vM « simple bipartite gruph G — (A\ y, f)

and di Mi-nroRf» 117
X-{* X,) ami Y-{y r, },
such that
</<!<*!)- 'I C# — 1.2 />).
<W*i 0-1.2 q).
if, and only if,
k k
A) I '•; > £s, {k- 1.2 ,/- 1),
B) tlt * " lf -,
Clearly, a necessary and sirfTicicnt condition that there exists such a graph
C is that there exists a I-graph whose dcmi-dcgrccs arc given by the pnirs
(()..*,). @. s2) @. sn) (r,. $,,,}. (/;. s,l+1) {rp,Sqip).
where
From Corollary 2, Theorem I. such a I-graph exists if. and only if, we have
both
A') y r* > £ st (k ■ 1.2 </ + />), and
These conditions arc equivalent to conditions (I) und B) above.
Q.F.D.
Theorem 8. The numbers r/|. </j dn constitute the degrees of a tree if. and
only if,
A) </, > 1 (f - 1.2 n),
0) X d, -2(h - 1).
From Theorem A9. Ch. 3). these conditions arc equivalent to
7T(ii ; tf,, «/„ ...,</„) 0.
Q.E.D.
•heorcm 9. Let dx > da >•••> f/B Ae a sequence oj integers, n > 2. /I

118 URAPHS
necessary and sufficient condition for the txistence of a simple connected gra
(i with degrees da(x,) - d,, is that
A) (/„>!
B) £</,>2(ii- 1)
C) ]T di
j-i
D) £ d>* £d, (k- 1.2 n).
J-I J-I
Suppose these conditions are satisfied,, then from conditions C) mid D) the
exists u simple graph with the given degrees, f-'rom B). it has at least n |
edges, and therefore, if this graph is not connected, it has a cycle (Theorem f
Ch. 2). Let [.v. y] be an edge of this cycle, and let [a, h) be an edge of a dillcre
connected component. This exists, from condition (I).
If edges [.v, v] and [a, /»] arc replaced by two new edges [.v, a] and [y, /;]. the
number of connected components is reduced without changing any degrees
My repeating this operation as many times as needed, we obtain a connect
graph.
Conrcrse/y. suppose there exists a simple connected graph G with degrees
dti(x,) - d,.
Then conditions A). C) and D) arc satisfied. Furthermore, if m denotes the
number of edges in G,
2 m > 2(m - t)
(since the cyclom«itic number satisfies »■«■#) -»»-«+ 1 > 0). Thus con-
condition B) is also satisfied.
Q.F.U.
Recall th.it a graph of order n > 2 is defined to be 2-connectcd if it is coil'
ncctcd and if it has no articulation vertex (a vertex whose removal disconnect
the graph),
Theorem 10. Let d, > d3 > ••■ > dA be a sequence of integers, n > 2. A
necessary and sufficient condition for the existence of a simple 2-tonnecK
graph G with degrees dj.x,) - d, is that
(I) dn > 2

DFGRITiS AND 01MI-DEORI EJ 119
n
0 Y. di > 2<" + <*i ~ 2)
<3) if,
*
D) I d, <£ d, (k - 1.2 «).
|. Assume that conditions A). B). C) andD) «/r satisfied. Then there exists
from Theorem 9 a simple connected graph G with d(,(xt) - </, for all i. If xk is
an articulation point, the subgraph G' generated by X - { xk) has p' > 2
connected components. At least one of these connected components has a
cycle, since
v(C') « m' - «' + p' - mi - dk - (n - 1) + p' >
1
X 4 - </, - « + I + p > p
-i-i
t > I
Let (>', z] be an edge of this cycle, and let [t, u] be an edge of another con-
connected component of C (which exists from condition (I)). If the edges |>\:]
and [t. u] .ire removed, and two new edges [y, i) and [:, u] arc added, the
degrees iire not altered, hut the number of connected components of G' is
reduced. Furthermore, for each vertex .v. this operation docs not increase the
number of connected components of the subgraph Gx-ui (supposing that
('.") has been selected in a cycle if its component in G' is not a tree).
Repeating this operation as many times as needed, we obtain a graph //
such that //x-ui has only one connected component (for each vertex x).
Graph // is the required graph.
2. Conrerseiy. the degrees of a 2-canm'cled graph G satisfy the required coih
ditlans, because Gx.lxii has /» — dt edges, n — 1 vertices, and
0 < \(GX. ,„,) - m - n' + 1 - in - tit - » + 2.
Thus
X d, - 2 m > 2(/t + d, - 2) .
Hence, condition B) holds.
Conditions A), C) and D) dourly hold.
Q.E.D.

120 ORAPHS
EXERCISES
1. For a inultigraph C wilhoul multiple edges (but possibly with loops) lei *<?(*) - d
if vertex x has no loops. Lei to(x) ■ du(.x) - I if vertex x has a loop. In other word ,
loop increases by I (noi 2) the "corrected degree" 8<X.v).
Show from Theorem 2 thai for a sequence rf, > d3 > — > </.. there exists emulligrapJi
C (wilhoul multiple edges) with n vertices xux3, .... x., such that 80(*i) ■ </■ for /
I, 2 it, if and only if
££ (A-t.2
1-1 l-l
(Ramachendra Rao (t969
2. Show that the pairs (r,, s,) ere the demi-degrccs of an arborcsccncc with root *i if, and
only if,
r,-0
ft - I (/ t)
1-1
3. Show that the pairs (/■,, j,) ere the demi-degrees of a strongly connected functional
greph If, end only If,
r,-51-t (/- I. 2,....«).
4. Show that a sequence rfi, </j,..., rfn constitutes the degrees of a multigraph if, and only
if. 2 d, is even. (Senior [t9SI
5. A simple graph G Is said to be k-edge-conntcitdif it is not disconnected by the removal
of less than k edges. Show that if n > I and k > I. then e sequence 4i. </a. •• . </» con-
constitutes the degrees of a A-edge-connected graph if, and only if,
A) the d, are the degrees of a simple graph,
B) di » k for all /. (J. Edmonds A9641)
6. Let d\ > (it >•••> rf« be a sequence of integers with 2 </, even. Shew that there exists
a mulligraph C wilh muliiplicily p wilhoul loops wiih vertices r, of degree </<,(v,) - </■ Ifi
end only If,
< mini J] d, +pkl — pk\
£ J] p -I. 2 «).
Hint: Thin can be shown from the Corollury io Theorem S.
7. Lei ((, ></,>•••></„ be a sequence of Integers. From Its corrected conjugnM
sequence (r7i,r73.di....).
Show thai <7| > <73 > G3 >—, or that there exists an integer A such Mint
<7i > dt> — > ^».
dm di, I > <7*.i > 7*.j > ••'.
8. Let rfi > </a >••• be a decreasing sequence, and let (<?,) be its corrected conjugate
sequence. Consider an integer k < ;/, end let A, be the largest integer / such that the num-
number of hatched or dotted squares in the /-th column of the corrected Ferrers diagram U
> A.

DEGREES AND DFM1-DE0RF.E* 121
For /o > *« *how lhnt
» / " \ "
£«fi-.min( £ * + */-<■]- £ d, ■■ klu k
l-l 1>* V-lt I / J-Jo+ I
Let ok denote Ihe number of empty squares In the square @. A] x [0. A) of the corrected
Ferrers diagram. For /« < A, show that
£ rf, minf' £ rf,+ */_
1-1 1>* M-l+1
9. Show thai if C is a 3-connecied graph with degrees dt < rfB <-"< </■,, then
A) A>3
B) £d >2(n-4+ /. +</„-!)
i
(J) £ rfi is even
i
D) £«/, < £rt.
1-1 l-l
(S. H. Rno and A. Kamachandra Kno [l%9] have shown that these necessary con-
conditions arc also sufficient for the existence of a 3-connecicd graph with degrees </,.)
10. Let </| > <l3 > — > rf» be a sequence of inicgen wiih I rf, even. Show ihnl there
nisisnmultigraph Cwiih nniltipllcily/> without loops with vertices x, of degree rfi(.v,) ■
rf, if. und only if.
rf, < />A(fc - I) + t min {pk, <t.) <* - 1. 2 «)
(V. Chungphaisan)
'I. Show, for ihe case k ■ I. that there exists a simple graph with degree sequence
Wi,... <U) contuining a regulur pnriinl graph of degree k if. and only if,
|i) I here Is a simple gruph with degree sequence \dx, ... <!„),
(■i) ihcre is a simple graph with degree sequence UU — k. .. , rfn — k).
'V. Chungphaisan hus verified this coniecitire of R. Grunbaum und S. Kundu has shown
■tai the above siaicmcnt is I rue for all k.)

CHAPTER 7
Match ings
I. The maximum matching problem
Given a simple graph G ■ (X, £), a matching is defined to be a set £0 < r
edges such that no two edges of £0 are adjacent. If £0 is a matching, and I'
£, c £0, then £, is also a matching.
We shall study the following problem: Find a matching Eo such that | £01
maximum.
A vertex x is said to be saturated by a matching £0 if an edge of £0 is at>
t ached to x. Let S(E0) denote the set of all saturated vertices. A matching that
saturates all vertices of G is called a perfect matching. Clearly, a perf .
matching is a maximum nuitching. In this chapter, we shall use dark lines 11
denote the edges of £0, and light lines to denote the edges of £ - £0.
Truncated chessboard
Fl«. 7.1
Maximum mulching of the
corresponding graph
liXAMl'Lii 1. Problem of the truncated chessboard. Consider an 8x8
chessboard whose upper left and lower right corner squares have been
removed. (See Fig. 7.1.) We have 31 dominoes, each domino covcrin
exactly two adjacent squares of the chessboard. Can we cover the 62 squa
of the chessboard with the 31 dominoes?
This problem is equivalent to finding a maximum matching in a graph
whose vertices correspond to the squares of the truncated chessboard. In thi
122

MATCHINOS 123
ph. two vertices arc adjacent if they represent adjacent squares in the chess-
board. (See Fig. 7.1.) It is easy to sec that the matching in Fig. 7.1 is not per-
perfect. A simple argument that does not use mulching theory shows thnt no
perfect matching is possible: Colour the squares of the chessboard black and
white (as usual). Note thnt the truncated chessboard docs not have the snmc
number of black and white squares, since the two missing squares must have
the same colour. Clearly, each arrangement of the dominoes covers the same
number of black and white squares. Hurcc, no perfect matching can exist.
If the chessboard had the same number of black and white squares, the
number of dominoes needed to cover it would be difficult to calculate without
using matching theory.
Example 2. The Battle of Britain. In 1941, the Royal Air Force consisted
of planes requiring 'wo pilots. However, certain pilots could not fly together
because of language differences or training deficiencies. Given these restric-
restrictions, what is the greatest number of planes that can be airborne simulta-
simultaneously? This problem is solved by finding the maximum matching in a
graph whose vertices correspond to the pilots and whose edges join pilots
who can fly together.
Uxamflu 3. Personnel assignment problem. An office has p secretaries
*,, x , xf and q jobs y,, y3 >■„. Each secretary is trained to perform
at least one job. Is it possible to assign each secretary to a job for which she is
qualified? Let /'(.v,) denote the set of jobs for which secretary .v, is qualified.
The problem reduces to finding a matching that saturates all the vertices of A'
in a bipartite graph (X, Y, F).
Fxample 4. Dating problem. In a co-educational American college, each
girl has A boy-friends, and each boy has k girl-friends. Is it possible to have a
dance in which all students simultaneously dance with one of their friends?
Later, it will be shown that this is possible.
Consider a matching Eo. An alternating chain is defined as a simple chain
('.c. a chain that docs not use the same edge twice) whose edges arc alter-
alternately in Eo and in Fo - £ - £0, i.e. alternating dark and light lines.
Lemma. Let G - (X, £") be a simple graph, and let £0 and L\ be two
Patchings in G, Consider the partial graph G' with edge set
(£„ - £,) u (£, - £0)
h connected component o/G' is of one of the following types:
Type I. Isolated vertex.
Type 2. Fven elementary cycle whose edges are alternately in Lo and Ei.

124 ORAPHS
Type 3. Elementary chain whose edges are alternately in Eo and Ex and who
cndpoinis are distinct and are both unsaiuraied in one of the %
matching*.
Let a € A'. We have three cases:
Casl 1. If a* S(E0 - Ei) and a$ S(£, - Eo). then a is an isolated ve
tcx.
Casf 2. If a 6 S(E0 - E,) and a S(E{ - Eo). then a is the endpoint of an
edge in Eo - Ex. No other edge of Eo - Ex is attached to a (because EQ \
matching); no edge of E, - Eo is attached to a (because a$ S(EX - £"„)),
Furthermore, a$S(Ex) (because, otherwise, an edge of F, attached to
would belong to E, - £0).
Case 3. If aeS(E0 - Ex) and aeS(E, - £„), there exists a unique
edge of Eo - Ex attached to a and a unique edge of Ex - £0 attached to .
Since these three cases arc exhaustive, the maximum degree of the pnrtial
graph
(A\(E0 £,)u(E,-E0))
is 2. This shows thiil the connected components must be of one of the types
described above.
Q.E.D.
Theorem I (Merge [1957]). A matching Eo Is maximum if. and only if, there
exists no alternating chain between any two distinct umtaturated rvrtkes.
1. If Eo is u matching for which there exists an iilternating chain between
two unsaiumicd vertices, then by interchanging the dark and light edges
along this chain, we obtain a new mulching Ex with | £, | ■ j£0| + 1.
Thus, matching £0 was not maximum.
2. Suppose mulching £u satisfies the condition or the theorem, and
let £, be u nuxiiiium matching. From 1. we know that mulching Ex satisfies
the condition of the theorem. Thus, | Fo - £, | - | £, - £01 since the
clementiiry chains of the partial gniph {X.(L0- £,)u(E, - £0)) ai"e
necessarily even. Thus
I£ol-I£, I.
and. therefore, Eo is a maximum mulching.
Q.E.D.
Corollary I. Let Fo he a maximum matching and consider the alternating
chain ft - (ex,fx, eu.fa, ...). where e, e Eo and f,e Fo - E - £0. Let the

MATCHINOS 125
operation which interchanges dark and light edges in p be called a "transfer" on
Each maximum matching Ex can he obtained from Eo by a sequence of trans-
transfers along rerteX'disjomt alternating chains that are, for the matching £0,
either alternating elementary cycles or alternating elementary even chains
starting at an unsaturated vertex.
It suffices to make the transfers ulong the connected components of the
partial graph generated by (£„ - £i)u(£, - £0), since the connected
components arc of the ubove two types by virtue of the lemma.
Q.E.D.
Corollary 2. An edge Is called "free" if it belongs to a maximum matching
but does not belong to all maximum mulchings. An edge e Is free if, and only If,
for an arbitrary maximum matching Eo, edge e belongs to an eren alternating
chain beginning at an unsaturated vertex or w an alternating cycle.
If c belongs to an alternating chain of this type, then clearly, e is free.
Conversely, iff is free, suppose, for example, that ps£0 and e$ Ex lor
sonic maximum mulching £,. Thus ee(E0 — Ei)u (Ex — Eo) and e
belongs to a connected component of the partial graph generated by
{£„ - £i)u(E, - Eo). Hence, for the mulching £0, e belongs to an even
alternating chain beginning at an unsaturutcd vertex or to an alternating
cycle.
Q.E.D.
Theorem 2 (FrdOs, Gallai [1959]). The maximum number of -dges in a
simple graph of order n with a maximum matching ofq edges (« > 2 q > 0) is
(V)
// n m 2 q ;
Note that in the case w ■ 2q, the graph Aa,, a clique with 2<y vertices.
ls clearly a graph of order n with m ■ ( ''j edges und with a maximum
"latching of cardinality q.
1" the second cusc, the gruph formed by the union of a {2q + l)-clic]iic

126 ORAPIIS
,., and a set 5n.(j<J*,) of/i - Bq + 1) isolated vertices is clearly a graph
2
of ordern with a maximum matching of cardinality q and with m ■ (
edges.
Finally, in the third case, take a ^-clique Kv and a stable set 5n_, and jo:n
in all possible ways the vertices of Kt with the vertices of Sn.Q.
Clearly, this graph of order n has a maximum matching of cardinality
since n - q > q and has »» ■ u) + 9(" - ?) edges.
We shall now show that the given numbers represent the maximum
possible number of edges. For the first case (w ■ 2 q). this is evident.
Suppose that
n>2q+ 1 .
Let S(E0) denote the set of unsaturated vertices in a maximum matching E
where | Eo | - q. Since n > 2 q, we have £(£0) 0- Let E, denote the
edges in matching Eo that have one endpoint adjacent to several vertices <
S(E0). By Theorem 1. the other endpoint of such an edge cannot be adjacent
to S(E0). because then there would exist an alternating chain between two
distinct unsaturated vertices, and Eo would not be a maximum matching.
Let £a - £0 - E,. and let 9, - | E, | and q3 ■ | £a |. Thus, q, + q2 ■ q.
For / - I. 2. let A', denote the set of endpoints of the edges of £"t. Thus.
Af.n-Yj-0. XinSiEJ- 0. X, n S(£o) - 0 .
1. Two edges of £, cannot generate a 4-cliquc because then there would be
an alternating chain joining two vertices of £(£0). Thus, the number of edges
of G joining two vertices of Xx satisfies
2. The number of edges of G joining A', and S(E0) satisfies
3. Let [a*3. ya) be an edge of E%. If neither xa nor ya is adjacent to the set
A7 of vertices of A", that are non-adjacent to S(E0). then
'"<;({ A":. y2 }. A- - A',) < 2 r/, + 2 < 3 g, + 2 .
If the edge [xa.ya] has an endpoint .va adjacent to A',', (sec Fig. 7.2). its
other endpoint ya is not adjacent to S(E0). Similarly, endpoint ya cannot be
adjacent to two vertices of A". Thus,

MATCHING*
127
Xi.X - Xx) < 2?, + I ,
mdv,.X- A'j) < v, + I.
and. hence,
Finally, we obtain
. X -
2</2
UK. 7.2
Consequently,
iw - /«„(*,, X - -V,)
. S(£o))
2
-2,)+B •')-
If n < g * . then «-
* * B)

128
or
GRAPHS
m
Combining the above inequalities Tor all cases yields,
m <« max
Note that
is equivalent to
or
2 q{2 q + 1) - q{q - 1) + 2 ql > 2 qn.
Q.E.D.
Miiximum matching algorithm
Consider a simple graph G - (X, E) with a matching £0. and associate
with it a I-graph G ■ (X, V), where (x,y) e V if there exists a vertex such
that [.v,:] 6 E - Eo and [:,;'] e Eo. For each even nitcrnating chain )t in C,
7.3

MATC1I1NUS 129
that starts from an unsaturatcd vertex, there corrcspoiuls in C a unique ele-
elementary path p. For example in lig. 7.3,
H - \a,b.c,e,il] corresponds to ft ■ [a. c. </] .
For each elementary path ft in C there corresponds a unique even chain ><
of 0'. but this chain is not necessarily simple. I-or example,
/i - [«, c. c/,/>] corresponds to n - [fl, h,c.e.</. c. b]
(edge /be appears twice).
A path in £ will be called legal if it corrcspoiuls to a single chain in 0'.
Otherwise, it will be called illegal.
Hence, the matching problem reduces to finding a legal path in G that con-
connects an unsaturatcd vertex (e.g. vertex a) to the neighbours or another
unsaturatcd point (e.g. vertices b or f>). One could use known algorithms Tor
finding all elementary paths in G (sec Ch. 4. § 1) and then each path of ft that
is illegal could be eliminated. In Fig. 7.3, the chain ft - [a. c, i>. g] yields the
desired alternating chain ft - [a, b. c. d. ej\ g, AJ in G,
As noted by Jack Edmonds ([1962], [1965]). it is not necessary to explore
each elementary path of 17 starting nt vertex a in order to reach an un-
unsaturatcd vertex by a legal path. Modifications of Edmonds' algorithm have
been suggested (C. Witzgall and C T. 7.ihn [1965]: M. Dalinski [1970]; I),
Roy [1969]).
2. The minimum covering problem
Given a simple graph C - (X, £). a coi-erinft is defined to be a family
fc t sUch that each vertex .v e X is the endpoint of at least one edge of F.
The problem of finding a minimum cardinality covering has many
similarities to the maximum matching problem. The covering problem is a
more general case of a problem known in logic as "Quinc's Problem".
Example. In the fort shown below (Fig. 7.4), there is a tower at the midpoints
I''i|l. 7.4. Minimum covering or a graph (in dark linn)

130 OR API IS
of each wall. A guard stationed at a wall can watch both towers at the end o
his wull. What is the minimum number of guards needed to watch all the
towers? Since the minimum covering of the corresponding graph is 7 edge
it follows that 7 guards will be required.
Theorem 3 (Norman. Rabin [1959)). In a simple graph G - (X, L')oforde
n, a maximum matching £0 and a minimum covering Fo satisfy
\E0\ + \F0\-n.
Given a maximum matching Eo, a minimum covering
is obtained by adding to E0,for each unsa titrated vertex y, an edge ev ofG that
incident to y. Given a minimum covering Fo, a maximum matching Et Is ob-
obtained by removing successively from Fo edges that are adjacent to an unremoved
edge.
If Eo is a maximum matching, the set
F, -E0Kj{eylyeS(E0)}
is clearly a covering, and
I /', I - I Eo I + (" - 2 | £0 I) - « - | £0 |
Furthermore, if Fo is a minimum covering, the set £\ obtained by the sue*
cessivc elimination of edges of Fo that arc adjacent to an (unreinoved) edge of
/•'o is a matching. Since in G, the edge of Fo do not form chains of length 3,
each removed edge creates exactly one unsaturatcd vertex of Ex. Hence,
\FO\-\EX\ -|A--S(£,)| -»-2|£,|
and
I Fo | - „ - | £, |.
Since | £j | < | £0 |. it follows that
\F, | - # i - | f0 | < fi - | £, | -\F0\.
Thus, the covering £, is also a minimum covering. Since | F, | - | Fo |. '•
follows that | £, | - | £0 |, and, consequently, the matching E, is a maximum
matching.
Finally, | £0 | + | Fo \ - n.
Q.E.D.
This theorem shows that the minimum covering problem reduces to the
maximum matching problem, which we shall study in the next section.

MATCHING* 131
^ Matchinjjs in bipartite graphs
A graph G is said to be bipartite if its vertex set can be partitioned into two
classes such that no two adjacent vertices belong to the same class.
Ihcorcin 4. For a graph G, the following conditions tire equivalent:
(|) G is bipartite,
B) G possesses no elementary cycles of odd length,
C) G possesses no cycles of odtl length.
A) B) because if C is bipartite, we can colour the vertices red and blue such
that two adjacent vertices have different colours. If G has an elemen-
elementary cycle of odd length, then the vertices of the cycle cannot alter-
alternate in colour.
B) C) Suppose that G possesses no elementary cycles of odd length, but
there exists a cycle /i - [.v0, a'i -vr - .v0] of odd length. If there
arc two vertices x, and xk in cycle n such that / < k and x, — xk,
then the cycle can be decomposed into two cycles n[x,, a\] and
n[x0. x,] + /i[xk, .v0). Furthermore, one of these cycles has. odd
length (otherwise, /i would have even length).
Clearly, each time that the cycle /< is decomposed in this way, an
odd cycle remains. When this decomposition terminates, there will
remain an odd elementary cycle, which contradicts B).
C) (I) We shall show that a graph without odd cycles is bipartite. Suppose
that the graph is connected (otherwise, each connected component
could be considered separately). Successively, colour the vertices
using the following rules:
Rule I. Colour an arbitrary vertex a blue.
Rule 2. If vertex x is blue, colour red all vertices adjacent to x. If
vertex y is red, colour blue all vertices adjacent to y.
Since the graph is connected, each vertex is coloured. A vertex x
cannot be coloured both red and blue, since then vertices x and a
would be contained in a cycle of odd length. This colouring deter-
determines u partition of the vertices into two classes and C is bipartite.
Q.E.D.
Henceforth, a bipartite graph with vertex sets X and Y and with edge set £
w'll be denoted by G - (X. Y, £). For any A c X u Y. the set of vertices
adjacent to set A is denoted by r0M).

132
GRAPHS
Konig's theorem [1931]. For a bipartite graph G - (X, Y, £), the maxima
number of edges in a matching equals
min
A c X
X - A
Consider the transportation network with vertices X\j Y and a source a ar j
a sink b. Source a is joined to each y,e Y by an arc of capacity c(a, y,) m ^
Sink b is joined to each x, s X by an arc of capacity c(x,, b) - I. Finally,
is joined to x, by an arc of capacity 1 if y, ra(x,).
Flu. 7.5
For a set A c X, the total demand of set A equals </(/f) - | A |. The maxi-
maximum amount of flow that can be sent into A equals F(A) - | ra(/4) |.
A flow in this network defines a matching in the graph in which x, and
y, arc matched if a unit of flow traverses arc (yt, x,). Conversely, each matcn-
ing defines a flow.
The cardinality of a maximum matching is, therefore, cquul to the value of a
maximum flow between a and b. By Theorem B, Ch. S),
max | £0 | - d(X)
Co
min {F(A) - (HA)) -
A c X
min (
A c X
rti(A) | -\A\)-
min (| A' - A | + rC)(A)).
A c X
Q.E.D.

MATCHINOS 133
For a graph G, a transversal set JTis defined to be a set of vertices such that
each edge has at least one midpoint in T. An equivalent formulation of
Theorem is:
Corollary I. For a bipartite graph G, the maximum number of edges In a
patching equals the minimum number of vertices in a transrersal set.
Let £0 be a maximum matching, and let To be a minimum transversal set,
Clearly. | To | > | Eo | since To contains at least one endpoint of each edge
in i'c
rurthcrmorc, for each A c A', the set T - (X - A) u ro(/f) is a trans-
transversal set of G, and from Konig's theorem.
| £o I - min (| X - A \ + | r(i(A) |) > j To | -
A C»
Hence, | Eo | - [ To |.
Q.E.D.
For a graph G, a j/a/>/<> act S is dclincd to be a set of vertices such that no
edge has two distinct endnoints in S\ another formulation of Konig's theorem
is:
Corollary 2. For a bipartite graph G. the maximum number of vertices in a
stable set equals the minimum number of edges in a covering.
If Tc X denotes a transversal set, and if £0 denotes a matching, then,
from Corollary 1,
max | Eo | - min | T |,
'o T
If JTis a transversal set, its complement 5 - (X\J Y) - JTis a stable set.
If 5 is a stable set. its complement is a transversal set. Thus,
max | S | - | X | + | Y | - min \T\.
s r
Prom Theorem 3,
min | F \ - | X \ + | Y \ - max | £0 |.
r Co
Hence,
max | S | - min \F\.
s r
Q.E.D.
'fa bipartite graph G - {X, Y, E) has a matching that saturates all the
Alices of X, then we say that X can he matched Into Y. If this matching also
^turatcs all the vertices of Y. we say that X can be matched onto Y.

C4 GRAPHS
The following theorem is un easy consequence of the Kttnig theorem.
Theorem 5 (P. Mall [1934]: "Konig-Hull Theorem"). In a bipartite gra
C - (X, Y, E). X can be matched into Y if, ami only if,
| rG(A) | > \A I (AcX).
l-'rom the Konig Theorem. X can be matched into Y if. und only if,
| X | - max | Eo | - min (| X - A | + | lti(A) |).
Co A e*
This is equivalent to
min (| ru{A) \ - \ A |) - 0
A CX
or
| ru{A) | - \A | > 0 M c A-).
Q.E.D.
Corollury 1. In a bipartite multigraph C - (A\ K. E) with
\X\-p, \Y\-q,
index the vertices xt e X and y, e Y such that
</<;(*,) < dc(x,) < - < <{u(xp),
A sufficient condition that X can be matched into Y is that q > p and
*2 do(yt) (fc-2,3 /»).
Consider two subsets A <=■ X and A c y with /c and A - 1 clement
respectively, l-'rom the above inequulity. it follows that
»iu(A. Y) - £ dG(x) > X dc{x,) > X
I dc{y)-mt,(X.B).
Thus, the number of edges leaving A is strictly greater thun the number tT
edges entering B. Hence, F0(A) <$ B for each set B of k - 1 elements; con*
scquently | IQ(A) \> k -\ -\A\- \. Finally,
>\A\ (A<zX).
Therefore, A' can be matched into Y.
Q.E.D.

MATCHINOJ 135
Corollary 2. If In a bipartite inultigraph G ■ (A'. Y, E), we have
min dc(x) > max dc(y) and I Y | > | X |,
^f/i A' row be matched into Y.
Let
min dG(x) - </„ max dG(y) ■ </, .
Thus c/| > rfg, and by indexing the vertices as described above,
> (A - l)</, >
1) (* - 2, 3 />).
Therefore, X win be matched into Y.
Q.E,D.
Corollary 3. If G » (X, Y, E) is a bipartite multigraph with no Isolated
vertices and with \ Y \ > | X |, and such that for some reritx xt e X.
min dti(x) > max do(y).
x*X
then X can be matched into Y.
We may suppose that .V| has minimum degree (otherwise, the proof is
immediate). Then
- + d(i(xk) 2> d(l(x,) + {k - \)d, >
>(k - I) (/, > </,;(..',) + - + dc(yk.x).
Hence, X can be iniitchcd into Y.
Q.E.D, .
("orollary 4. In a bipartite multigraph G ■ (X. Y, E). ther> exists a match-
'"K that saturates all the vertices with maximum degree.
First, suppose that there exists a bipartite inultigraph
G -(X,Y.I),
*uh G m [X, Y. E) us a subgraph suppose X c JP. > c P. and G is regular
of degree
h ■ mux d(;(z).
from Corollary 2. JP can be matched onto ? in G. since | J| - | 7\.
nis matching saturates each vertex in G of degree h.

136 GRAPHS
We shall construct this graph G by taking h replicas of multigraph
Denote these h replicas by
G - {X, Y,E), C - (A", )".£'), G" - (A", )'",£")....
Let X consist of Xu X' u-u A/(ll"l> together with some addition 1
vertices. Similarly, let P consist of Vu 1" u-u y(l>-» together with some
additional vertices. These additional vertices are determined as follows:
x,e X and da(x{) < h. create, in F. It - <}0(x,) additional vertices, and jom
each of these vertices to x,e X, x',e X'.tfe X" x\*-"e A"""", th
analogues of x,. Repeat this construction ifj^e Vand da(y,) < h.
In this way, a multigraph G having G as a subgraph is constructed.
Q.E.D.
Note that this result allows us to give an affirmative answer to the Datin
Problem (Example 4. § I).
Corollary 5. In a bipartite graph G - (X, Y. E). X can be matched into Y
if. and only If,
\X-r<JLB)\*\Y-B\ (BczY).
A) If A'can be matched into Y, and if B c Y, then from Theorem 5,
| x - rc(B) |«; | rG{x - W)) \.
No vertex x 6 X - r^B) is adjacent to B. Thus,
rc(x) c Y - B,
and, hence.
| X - l(,(B) | < | ^(A- - rG(B)) \*\Y-B\.
B) If X cannot be matched into )'. there exists a set A <= X such th
| A | > | r^A) |. Let B - Y - F^A). Since no vertex of B is adjacent to
A c X - r(i{B) .
Hence.
| X - rc(B) | > \A | > | rc(A) | - | Y - B\.
Q.E.D.
This corollary is in fact a reformulation of the Ktinig-Hall theorem thai
will be needed later.
The next result is a reformulation of Bernstein's theorem.
Theorem 6. In a bipartite graph G - (X, Y. E), a necessary and sufficient

MATCHINOS
137
that there exists a matching that simultaneously saturates X and
0 Ylsihat
n) X can be matched into Y. l.e.
| rc{S) | > | S | (S c X);
B) B can be matched into X. i.e.
\rQ(T)\>\T\ (TcB).
Clearly, conditions (I) and B) are necessary. We shall show that if there
exists a matching £0 from B into A <=■ X, and that if there exists a matching £t
from A'into Y, then there exists a matching saturating both A'and B.
We shall now construct from E{ a matching E[ in which the saturated
vertices of X remain saturated, and an unsaturulcd vertex b e B in Ex becomes
saturated in E[. Since beS(h'o). b is the end-point of a chain ft of the form:
with
;. r] -
,... e£0 - £, ,
— £0 .
— Edge of E, £0
— Edge of F.o - Ht
I-Ir. 7.6
Suppose that ft is as long as possible. Then, the last vertex z of this chain
felungs to >' (otherwise, r belongs to A. and r is un endpoint of an edge of
fi - £0 that can extend chiin ft). If re >'. then r $ ^(otherwise r is an end-
P°'»i of an edge of £0 - £, that can extend chain ft). Thus, ze Y - B.
Thus, £,' - £', u (// n £0) - (/4 n £,) is a matching saturating b: besides,
each vertex of X u B that is saturated in £, remains saturated in E[.
By repeating this procedure as many tiincs as needed, a matching saturating
°°tli X and fl can be obtained.
Q.E.D.

138 GRAPHS
Corollary. In a bipartite graph G - (X, Y, E), a necessary and suffcl
condition that there exists a matching simultaneously saturating X and B
is that
Using Corollary 5 to Theorem 5, the two conditions of the above theorc
can be written as
(i) | r<;(S) | > I s I (s<=x)
B') I B - ro(S) I < I X - S I (S c X)
or
B") | X | - | B - ra(S) I > I 51 (S<=X)
Q.E.D.
This condition was obtained independently by linear programming method
by Hoffman and Kuhn [1956].
Ixtmma (Folkman, Fulkcrson [1967]). Let G - (X, Y, E) be a bipartite
multigraph i\iih maximum degr >e < h and with | E| - mi. Let mi', mi", h' and
h" be positive integers with
mi' + mi" ■ mi, h' + h" ■ h .
The edges of G can be partitioned into two classes £' and E' (that form /M>
partial multigraphs G' and G"), such that
| E' | ■» m , | E" | - mi" , max dc(z) < h', max rfo-(r) < A
//, o/if/ only If.
m'-hl\X-A\-h'\Y-B\< mo(A, B),
mi" - /i" | X - A\ - h"\ Y - B\ < mti(A, B),
for all A c X and for all B c Y.
1. Necessity. If there exist two partial multigraphs G' and G" satisfying
the above conditions, we have
mi' - | E' | < mu(X - A, Y) + mo.{X, Y - B) + mc.(A, B)
*h'\X-A\ + h'\Y-B\ + mo(A,B).
2. Sufficiency. Consider the bipartite transportation network R obtained
by joining each x, e X to a source a and by joining each y,e Y to a sink
and by adding a return art (h. a). Let
a, - max { 0, do(xt) - h"}, fij m max { 0, do{yj) - h"} ;

MATCHINGS 139
i*t the interval of permitted flow values in arc u be
i [«„/»'] if m-(o,x()
)lPj,h'] if u-(yJtb)
[b(u), c(u)] m j [0. mG(x,, >•,)] if u - (x,, >',)
![/»',/«'] if «-(/>,«).
Each flow in R that is compatible with these intervals determines a partial
graph C with | £' | - m' and with
max { 0, du(Xl) - h"} < du.(x,) < W
The partial graph G' - G - G' satisfies | E" \ - mi" and
Thus the required partition of the edges of 6* has been found. Conversely,
such a partition determines in graph R a compatible flow within the per-
permitted intervals. From the Compatible Flow Theorem (Ch. 5. § 2), a necessary
and sufficient condition for the existence of a compatible flow in R is that
J) < c((o*(S)) (S c X \j Yu { a, b }).
It is left to the reader to verify that the above condition is equivalent to
that of the lemma.
Q.E.D.
Ihcorcm 7. (Dulmagc, Mendelsohn A961]). For a bipartite multigraph
C ■ (X, >', F) with | £ | ■ m and with maximum degree </i, let
a m max { m - mu(A. B) - [It - 1)(| X - A | + | V - B |)} .
A A
Ac A
p - min { mti(A. B) + \X - A\ + \Y - B\\ .
ACS
»c »
Then,
Furthermore, for each integer m' such that a < mi1 < /», //icrp f.\7.vM a
"latching E' c £ u-iV/f | £' | - wi1 u-Aotr removal results In a multtgraph with
Maximum degree < A - I.

140 GRAPHS
1. If [£'] > p, then there exist sets A c A'and B c y such that
m > h{mc(A, B)+\X-A\ + \Y-B\)>
> mc(A. B) + mc(X - A, Y) + mc(A, Y - B)
which is a contradiction. Thus,
[?]«'■
2. If 1.1 < a, then there exist sets A c X and B c y such that
^ < »i - /*„(/<. B) - (h - I) (| A1 - A | + | y- B |).
Hence,
mi - y > mG.M. B) + (/» - I) (| X - A | + | y - 0 |) >
C' i -v -
- .4. y>
which is a contradiction.
C) If in' satisfies a < m' < p, then, for till /I c ^ and for all Be K, wv
have
| m' > m - iho-(^. B) - (/f - 1) (| ,Y - A | + | Y - B |)
I mi' < mb4A. B) + \X - 4\ + \Y - B\.
Let A' ■ 1, A" - A - 1, mi" ■ mi - mi'. The above inequalities become
( in" - h" \X-A\-h"\Y-B\* mti(A. B)
\ ,„' - W\ X - A | - *' | >' - B | < in(i[A. B).
From the lemma, G can be decomposed into two partial bipartite multi
graphs
C - {X. £') and C" - (X, E - E).
with
| £' | - m', max dti{z) < I. max i/(;-(r) < h - 1.
Clearly, E' is the required matching.
Q.E.D.

MATCMNOS 141
^ An extension of the KttniR theorem
In this section, we shall consider a multigraph with the properties:
A) there are no loops,
B) if ft and /(' are two odd cycles without a common vertex, there exist
two adjacent vertices xs/j and x' e //.
Such a graph is called seml-blpartite (see Ch. 6, § 2). We shall now study
conditions for which a scmi-bipartitc graph possesses a pcrfxt matching
(i.e. a matching that saturates all vertices).
Lemma 1. Let G - (X, F) be a symmetric semi-bipartite l-graph with [ X \
tren; there exists a perfect matching if, and only if, there exists a partial graph
llafG with
d;,(x)-dii(x)~\ (xeX).
The proof follows from Theorem E, Ch. 6).
lemma 2. Let C — (X, D be a l-graph; there exists a set of elementary
circuits ofC that partition X if, and only if,
\r(A)\>\A\ (A<zX).
Associate with 0" a bipartite graph Gx - (X, X, £) obtained by taking two
replicas A'and A7 of set A', and joining at, 6 A'to^e A7 by an edge if x, e F(x,).
In graph G, set A'can be partitioned into circuits if, and only if, the bi-
bipartite graph Cj has a perfect matching, i.e., if, and only if.
I rei(A) | - | HA) \>\A\ (AcX).
Q.E.D.
Theorem 8. A semi-bipartite graph G ■ (A', E) possesses a perfect nutch-
% if, and only if,
A) I A'| is even,
B) | ro(A) \> A (/l=n
Apply Lemmas I and 2 to the symmetric scmi-bipartitc l-graph G*
obtained from G by replacing each edge by two oppositely directed arcs.
Corollary. // G is a semi-bipartite regular multigraph of degree h that has
Q" wen number of vertices, then G possesses a perfect matching.
Make two replicas A'and A"" of the vertex set of G, and construct a bipartite
"Hiltigraph // - (A', Z E) with m,,(x,y) - mo(x,y). For each A c X, we
have
/( | A | - m,, ({A. r,,(A)) < m,, (X, r,,(A)) - h \

142 GRAPHS
Thus | rc(A) I > I A I and, from Theorem 8, the simple graph G' obtain
from G by collecting the multiple edges has a perfect matching, which is alt
a perfect matching in G.
QE.D.
5. Counting perfect matchings
For certain graphs (particularly planar graphs) simple methods arc avai
able to count the number of distinct matchings. These methods arc the rest t
of work done independently by Kishcr [1961] and by Kustclcyn [1961],
Kastclcyn's proof has been simplified by H<Snyi [1966] and by Pla [1970). We
shall use Pla's proof.
First, let us review some definitions from matrix algebra. If A — ((a))
square matrix of order n, the determinant of A is written us
Det A -
where
/ 1 2 ... „ j
WO <tB) ... a{n)J
is a pcrmutution of degree n, and e(o) - + I or - I if the permutation "
even or odd, respectively. The permanent of A is defined to be the number
Perm A - £ flj(,,«j(j,... <C,.
e
Proposition I. Let G ■ (X. V) he a /-graph with vertices .vJt x2,.. ,XT
and let A - ((«})) be a square matrix oforder n definedby a\ - I lf(x,,.\;)e U,
and a) - 0 if(x,, x,) $ U. Then the permanent of A equals the numbir of pair-
wise disjoint systems of circuits that partition X.
Each non-/cro term of the expansion of the permanent corresponds t
such a system of circuits, and conversely.
Q.E.D.
A skew-symmetric matrix I) - ((/>})) can be associated with an anti-
antisymmetric I-graph C - (X, U) by letting
(+1 if (x,,Xj)eU,
b'j- - I if (Xj,Xl)eU,
[ 0 otherwise.
Matrix B is called the adjoint matrix of C.
If n - ((/>})) is a skcw-syinmctric matrix of even order n - 2 k, th-
pfaffian of B is defined to be
Pf fl - I e(ff,Nj; *!;...*&-,

MATC'HINCIS 143
ft - ['i. 'a]« ['3. '«]■••['ak-i- 'a*] is " permutation of degree n that
decomposes into k cycles of length 2 (i.e., n(i) - j implies./ / and n{j) ■ i),
sUch thut
'l < fj.'i < '4 Ilk-I < '«.
and where on is the permutation
/I 2 ... »\
\'i lj • • • V
Proposition 2. //" 0' ■ (X, V) is an antisymmetric I-graph of even order
,,m 2 k. and if B - {(b1,)) is Its adjoint matrix, then | Pf B | w /ess than or
equal t > the number of perfect matchings in G. Furthermore, the number of per-
perfect matchings in G equals | Pf B | ;/, and only If, each term in the expansion of
Pf B has the same sign.
Each term in the expansion of the pfalfian corresponds to a perfect match-
matching, and conversely. If two non-zero terms have opposite signs, then they can-
cancel out one another.
Q.E.D.
Note that the pfalfian is easily calculated using the following well-known
theorem from linear algebra:
Proposition 3. If B is a square skew-symmetric matrix of order n, then
Del B m (Pf BJ ifn is even,
m 0 ifn Is odd.
The proof can be found in most comprehensive texts on linear algebra.
Consider an anti-symmetric 1-graph 6* * (A, V) with a perfect matching
"'«<= V. If/< is a cycle of G, let n' denote the ares of u that arc directed in the
direction of travel through the cycle, and let | u* | denote the cardinality of
'his urc set. If, for a family M - {ft, / iel) of cycles, the numbers | fi,* | arc
*U odd. G is said to be well directed with M. Fiiuilly. cycle n is said to be
"litmahle if there exists a perfect matching in which yi is un alternating cycle.
theorem 9 (Kastclcyn [1961]). Let G « {X. U) be an anti-symmetric
Hraph of even order 2 k; let B be its adjoint matrix, and let Wo be a perfect
"Hitching in G. The following three statemints are equivalent:
A) All the mm-zero term in the expansion ofTf B have the same sign,
B) G is well directed for the family of its alternable cycles.
C) G is will directed with the family of the alternating cycles for Wo-

144 GRAPHS
If Wo is the only perfect matching in 0". the result follows since G has it
other alternable cycle. Thus we may assume that C has several pcrfec
matching;.
(I) B) For example, let /« - [v,, xa xap. .v,] be un alternable cycle
und let W and W be two perfect match ings in G such that
(W - W')\j{W - W) m ,,.
Suppose that the set of arcs common to it and W is
Wcs W-{<*„.,. *i,.i> <*«-,.*»)}.
The terms corresponding to W and W in the expansion of Pf 6 are
respectively
and
where
-c
2 ... 2k
2 ... 2 *
. l\ 2 3 ... 2p 2p+l ... 2k\
° "ll 2;> 2 ... 2 p - 1 2/) + I ... 2fc/-
Note that r.(o)c(o') - + 1, because 2p - 2 transpositions are
required to pass from a to a'. Thus, from (I).
+ | . off - -bl1bl...b\>-1 b\' •
Consequently,
\n+\m\tr\u\ (mod 2).
and for the alternable cycle ft, graph G is well directed.
B) => C) This follows because each alternating cycle of Wo is an altcrnable
cycle.
C) *• (I) Let Wbc a perfect matching different from Wo. From Corollary •
to Theorem 1, a sequence of perfect matchings Wo, W, M^can
be constructed so that for each / the arcs of
define an elementary cycle fi, that is alternating in tV0, Thus,
C). | ft,' | is odd.

MATCHING* 145
As above, we can show that the terms (H.W0) and fl(M ) corresponding to
matchings IV0 and W satisfy IKW^OOV) - +1. Consequently, (ill non-
itro terms in the expansion of Pf B have the sumc sign.
Q.E.D.
Theorem 10. // C - (A. U) is an anti-symmetric I-graph, and if M -
liiiP ') '5 " family af linearly independent cycles, then, by reversing the
direction of certain arcs, a \-graph C — (X, W) that is well directed for M can
bt obtained.
Let
s, ■ number of arcs of C directed in the direction of travel in /»,,
s' ■ number of arcs of C directed against the direction of travel in /*,.
Let
1-1 if
-0 if
Finully, let
(■ 1 if the direction of arc u, should remain unchanged
to obtain graph 6",
— 0 otherwise.
Graph C will be well directed for M if, and only if. for each /,
.ha, is °'ml (m0d2)"
m
"i + Z cu zi ■ ' (mod 2)-
Hence, C is obtained from a system of equations of n\ variables 2,, za
\ in the field of integers modulo 2:
m
T c,jZj m 1 + s, (lei).
If family M consists of independent cycles, then this system consists of
Principal equations and. therefore, it has a solution (z,, za zm) in the frcld
of integers modulo 2.
Q.E.D.
For example, consider the non-planar gruph in Fig. 7.7 with the perfect
pitching H'o shown in dark lines. The alternating cycles are linearly indepen-
since each possesses a distinct edge (marked with a cross in Fig. 7.8).
s, from Theorem 10, it is possible, by directing the edges, to obtuin a
C - {X, U) in which the alternating cycles are well directed. The
yjoint matrix of graph G - (A', U) is:

146
B
ORAPHS
0
- 1
- 1
- 1
0
- 1
1
0
- 1
1
0
1
1
I
0
- 1
0
- 1
1
1
1
0
1
0
0
0
0
1
0
- 1
1
1
!
1
1
0
From Flicorcni 9, the number of distinct perfect matchings is
| Pf fl I - v'Det Ii - v/3-3-2-2 - 6
We shall show that this method ill ways works for planar gruphs.
Theorem II (Kastclcyn [l%l]). Let G - (X, U) be a connected planar
graph with a perfect matching Wo thut is well directed for the family of con-
contours of its hounded faces. (From Theorem 10, one such orientation ainay
exists.) Then G Is well oriented for the family of Its alternating cycles.
Let ft he an ultcnniting cycle for the perfect matching Wo. Cycle ft
rounds an even number of vertices because the vertices in the interior /< art
matched together by Wo.
Let // be the subgraph of G generated by vertices situated on /< or in tM
interior of ft. Clearly, // is planar and connected. Suppose H has n vertices,
arcs and/finite faces, v,, va v,.
Successively traverse the cycles v,, va vr along their direction and then
traverse ft against its direction. While doing this, count the total number C
arcs traversed along their direction. It is clear thai each arc of // will
traversed once in each direction; thus, <J — m.

MATCHING* 147
Mcxt. by summing over the cycles, we have
i- I K'I+ ('</<>-I><*l
i-1
/<) is the length of/«.
Since G is well directed for the cycles vM we have
m ■ /+/(/*) + |/4+ |.
The numbers /;. m and /satisfy the [;uler relation for planar graphs (see
Corollary I. Theorem 2, Ch. 2)./- m - n + I. Hence
| n' | ■ hi +./' + Kft) m n + I + /(/«)
■ I + (/i - /(/<)) ■ I ,
since n - /(/') is the number of vertices in the interior of /i, which is even.
Thus, G is well directed for ft.
Q.E.D.
Remark. To calculate the number of distinct perfect matchings in 4 simple
planar graph G, it suffices to give the edges of G a suitable anti-symmetric
orientation, to determine the adjoint matrix B and to calculate \/Dct B.
EXERCISES
1. Suppose that G ■ (X, K) i* a connected gnph without iMhmi (nn edge whose rcnioval
disconnects the graph), and each vertex or G has degree 3. Consider 0 maximum matching
£>. Show that Micrc exists a chain whose edges belong alternately to £0 and F — /To that
uses exactly once each edge of £ - £0 and uses each edge of Eo twice.
(P. Medgycssy |I95O])
z- In a bipartite graph G - (X, Y. £). let
fl A) m 1 a I — I r,.iA) |. d0 max 6(A).
Show that ""*
,\jAt)i <5(/f| nA,)> MA,) MAj).
<2) Using the above inequality, show that the family
I A IMA) fiu)
» A\ «J Au
^O) IT t0 > 0, rhe set of vertices of X not saturated by at least one maximum matching
A,*/
y eh"*" 8 b'Parlitc Br«Ph C - (AT, K, /1) Tor which there exists a matching or X into
• Show that there exists an xn AT Mich that, forcich ,v /',(.vn). at lcist one maximum
m«ehmg uses the edge lx0. y\.

148 ORAPHS
Hint: If | r,AS) | > | S I for each S * a, nny jr0 AT can be chosen.
(M. Hall
4. Deduce from the preceding exercise that if the minimum degree tl<Ax) for x k
equal to k. then there exist at least k ! distinct maximum matchings.
(M. Hall [194 )
5. In a bipartite graph (X. Y. H with I X | - | Y |, let
k - max{ r/«(.vi) +• rfr.(jrj) + ••• +• daixn) — da(y\) — rfoO-j) — •••— Wh.|)} ,
Show that there exist k disjoint matchings of X into Y. (O. Ore [195
6. Show thnt in a graph G with n vertices and minimum degree k, a maximum marchir
V satisfies
„■—.(*.[;]).
(P. Erdos, L. Posa (I962T
7. Show thar if V Is a matching and T is a rranavcrsal set, then
mini T\ < 2max| V\.
Show that the equality holds if. and only If, the connected components of the graphs t,
nil cliques of odd cardinality.
8. If * - min do(x), if G is connected, and if max I V | < '-—^, then
mini 7*1 < 2 max I V\— k.
(Erdos. Callai [19611
9. If a graph remains connected after any k - I of its vertices are removed, and
max | V | < ^-^— , then k < max | V |. and
mini T\ < 2 max I V\ — k
(Erdos. Callai [I96I1
Show that this bound is attained by a graph G formed from a /('Clique Kk and /
k 4- I cliques K*,,, lt with each vertex of Jf«a,.i being joined to each vertex of K*.
10. Show that
2 min | V | < m + max I V |.
Also show by an induction on m that the equality holds if. and only if, the connect
components of the graph are 2-cliques or 3-cliques.
11. Show that
4 min | TI < 2 n + m — max I V \.
If C is a 2-clique. 3-clique. 4-cliquc or two triangles joined by one edge, then the cqualitf
holds. Arc there other connected graphs for which this equality holds?
(Erdos. Gallai |IWI»
12. Let a tree with diameter < 3 be called a double stor. Let/(C) denote the minimuf*
number of double stars needed to cover all the edges of a graph G with n vertices.
A) If r Is the cardinality of a maximum matching of C, show that
/(ft <2j-.
B) Show that /(C) < n - 2r.
C) Using (I) and B). show that /(ft < [^
(L. Lovasz [I

MATCIHNGS 149
,1 |n a simple graph C - iX, E). consider a family of sets
iCCC)
*fu i o I odd (/- 1.2 p).
B) For each [v. y\ e F. there exists an index I sui.li that
ICin{jr.y}| -min { I Ci |.l ! jr.r } l}.
,f I C. | < 2 * + I. Icl c(C) ■ max {*.!!. and let
- f <(G).
Show that for each matching £<,,
Alio show that
max ; Co | « min c({f).
£c
(J. Edmondx A964))
14. let C - (A*. >'. /*) be a bipartite I-graph without isolated vert ices (w here r'n a corre-
correspondence from AT onto Y). Show that the minimum number of arcs that must be added
to C to make a strongly connected graph Is max A*1.1 Y | ).
Htm: Use Corollary I to the Kttnig theorem.

CHAPTIiR 8
c-Matchings
1. The maximum c-matching problem
Consider a multigraph C - (X. £) with vertices .v,, xa xn and a
tuple c - (c,, ra <■») of integers with
0 < c, < </„(*,) (/-I, 2 «)•
The set £0 c £ is called a c-matching if for cacli /, the set £0(*i) of ed
of £0 incident to x, sutisfics
A vertex xt is suid to be saturated in the c-matching £0 if | £o(v,) | — c,.
In this section, we shall study the problem of constructing u maximum
c-matching. The maximum matching problem (Chap. 7) is u special cose o
the maximum c-matching problem for c - (I. I I).
To each multigruph G. there corresponds a simple graph G defined at
follows:
For each x, e X, define two disjoint sets
/!,- {aileeE(x,)} and B,-{6}/fc-|.2 dG(x,) - c,}
Let the vertex set of C be the union of Ur.i A, and of U."-i B,. For each /
join each vertex of A, to each vertex of B,. For each edge e - [x,, x,) of G,
construct an edge I — [a', a'] in G.
ITicorcm I. A maximum matching Eo in graph G that saturates L)P-i
induces a maximum t-matching Eo in graph G, and conversely.
I. Let J?o be a maximum matching of G. We may assume that Eo suturat
each /;[* (by interchanging if necessnry the edges of Eo and of E — Eo alofJ
a chain [bf, tf.aj] of length 2). Matching Eo in G defines a set of edges o
in G, and this set £0 satisfy
I Eq(*,) I < </«(*,) - I Bt I - Mxt) - (ddx,) - c,) - c,.
Thus, £0 is a c-matching in G; furthermore, '
I Eo I - I Eo | - £ | B, | .
150

C-MATCHINOS
131
2 Now consider a maximum c-mutching f, c E of graph G. Set
defines in G a mntching Et that saturates each /;J\ as shown in Fig. 8.1.
■E -/■„
Fig. 8.1
Consequently,
I E, I - I Ei I + I I «. I •
el
Since Eo is a c-matching in C, | Eo | < | E, |, nnd therefore
I f. I - I Ei I + 11 B, | > | Eo I + 11 «j I - I Eo I -
Since Eo is a maximum matching. | ., | - | To |. Consequently. | E, \ -
l^o I- Hence, £0 is u muximum c-matching of G, and £, is a maximum
watching of G.
Q.l-.D.
Remark. This theorem demonstrates that a maximum c-matching can be
instructed by determining the maximum mutching ^0 in graph G and then
^'"rating cuch vertex />? by an interchange along alternating chains of length
Consider a multigruph G with a c-matching £0. I-rom G, construct a
"digraph R(E0) by adding to G a vertex .v0 cnllcd the origin that is joined
0 cach vertex x, by r, edges, for all i. Multigrnph R(E0) is called a transfer

152 or Arm
Let the edges or £0 be represented by dark lines, and let c, - t
edges from x0 to x, Tor all / be also represented by dark lines. All other ed
of R(E0) arc represented by light lines.
An alternating chain is defined as a chain of R(E0) with edges alternate
dark and light that repeats no edge. A transfer along an alternating chain
is defined as the interchange of the dark and light colouring along n.
Theorem 2. (Berge [1958]). A c-matching Eo of a multigraph G - (X, £)(,
maximum if, and only if there exists no alternating chain in R(E0) that joi'
x0 to itself and has dark initial and terminal edges.
As in Theorem I. construct a graph G corresponding to the multigraph •
and a matching Eo in G corresponding to Eo that saturates each vertex in
U B,. An alternating chain with the above properties in R(E0) corresponds i
G to an alternating chain that connects two distinct unsaturatcd vertices
(if Eo is properly chosen), and vice versa. In graph G, such a chain exists
if. and only if, the matching Eo in G is not maximum (Chapter 7,>Thcorcm I),
or. from Theorem I, if, and only if. the c-matching £0 in G is not maximun.
Q.E.D.
Theorem 3. // Eo and E, are two maximum c-matchings of a multigraph
G - (X, £), then E, can be obtained from Eo by transfers along alternating
cycles ofR(E0) that arc pairwise edge-disjoint (but not necessarily elementary).
From Corollary I to Theorem I (Ch, 7). each maximum matching i
in Ccan be obtained from the maximum matching £0 by a series of transfer!
along vertex disjoint alternating chains. Euch of these chains is cither an
alternating elementary cycle or an even clcmcntury chain starting at an
unsaturatcd vertex. In cither case, this alternating chuin corresponds in
R{E0) to an altcrnuting cycle that is not necessarily elementary.
Q.E.D.
Theorem 4. A "free edge" is defined to be any edge ofG-(X,E) that
contained in some maximum c-matching but not in every maximum c-matching.
An edge e is free if, and only if, given a maximum c-matching £0 of G,
e lies on an alternating cycle of R(E0).
1. If e is contained in an alternating cycle of R(E0), then e is evidently
free edge, since a transfer can be made along this alternating cycle.
2, Let c be u free edge, and suppose at first that e is contained in the maxi-
maximum c-matching Eo- There exists a maximum c-mutching £, that does not
contain e. Thus, from Theorem 3, e is contained in an alternating cycle of
R(E0).
If e is not contained in £0, then e is contained in a maximum c-match*

C-MATCMNO8 153
p obtained from £0 by a transfer along an alternating cycle of R(E0). Again.
. js contained in an alternating cycle of R(E0).
Q.E.D.
2. Transfers
In this section we shall consider a simple graph G and show how to ob-
obtain from G all the graphs with the same degrees as G by a sequence of trans-
transfers of a particular form.
Lemma. If G - (X. E) is a simple graph such that each even cycle of
length > 4 has a chord that divides the cycle into two even cycles. Let Eo c £
be a set of dark edges. Then any transfer along an alternating cycle n can be
obtained by a sequence of transfers along alternating cycles of length 4.
If the length of cycle (i equals 4. the result is trivial. If the result is true
for cycles of length < 2 k, then we shall show that it is also true for un
alternating cycle n of length 2 k, say
ft - [a,,a2 au,a,].
Since this cycle has a chord of the form [al,au.ani]> we have two even
cycles:
H, - [alt a,+ alt2p*i, ot),
Vi ■ [a Of °/+j|i+i. ol+2f*2 o2k. fl|} •
Relative to Eo, only one of these two cycles is alternating, say ft,. Since
the length of /(, is < 2 k. the transfer along n, can be accomplished by a
sequence of transfers along alternating cycles of length 4:
£0 -» £o - £0 - (Hi n £0) u inx - £0) .
Relative to £0. cycle y«a is an ultcrnating cycle of length < 2 k. The transfer
r0 -» £o - £i - (/ii n E'o) u (n, - E'o)
can be accomplished by a sequence of transfers along alternating cycles of
•cngth 4. Thus, the transfer
£0 -» E"o - £0 - (n n £0) u (» - £0)
h°s been obtained.
Q.E.D.
"theorem 5. Let //0 - (A'. £0) and //, - (X, £,) be two simple graphs, with
'** same vertex set, such that
d,K (x,) - dHl (.v.) - d, (i - 1, 2 n) .

154 ORAPHS
Let a. b, c. d be any four vertices in X with ace Eo, bde Eo. ad$ Eo
bci £0. A "direct transfer on Ho" is defined as an operation that rentoi
edges ac and bd and adds edges ad and be. Then, graph //, can be obtainedfr
graph //(, by a sequence of direct transfers.
Clearly, Ho and //, are two partial graphs of a complete graph of the
G - (X. E). In G each even cycle [at. oa fli] has a chord of the
[a,, a, + 3]. From the above lemma and from Theorem 3, //0 can be tran
formed into //, by transfers along alternating cycles of length 4. These tran
fers are direct transfers.
Q.E.D.
Theorem 6. Let //„ - (X, Y, Eo) and //, - (X, Y, £,) be two bipartitt
graphs such that
- dUl(x,) - r, (/ - 1, 2 p),
d,,0(yj) - dlh{.Vj) - s, {j-\, 2 q).
Let Xi,xa X he two distinct vertices, and let y,,ya€ Y he two distin 1
vertices with
x,y,eE0, x2y2eE0, x,y24E0, x2y{iE0.
A "bipartite transfer on ll0" is defined to be an operation that removes edges
xi >'i and x2 yt and adds edges xx y.u and xt yt.
Graph //, can be obtained from graph //0 by a sequence of bipartite trans'
fers.
Clearly, Ho and //, arc partial graphs of a complete bipartite graph of
the form C - {X, Y, E) in which each xe X is joined to each ye Y. From
the lemma and from Theorem 3, Ho can be transformed into //, by a sequence
of transfers along alternating cycles of length 4. These transfers are bipartite
transfers.
Q.E.D.
I hcorcm 7. Let G - {X, U) and G' - (X, U) be /-graphs such that
<lc(x,) - da-{x,) - r, (/ - I, 2 «).
-W ■ Sj (' ■ I. 2 «)•
Let a. b, x, y be vertices of X with a b. x y, (a.x)e U, {y
{a,y)$ V, (b,x)i U. An ''oriented transfer" is defined to be an operatic
that removes arcs (a. x) and (/>, >') and adds ares {a, y) and (/>, x).
Graph G' can be obtained from Graph G by a sequence of oriented transfer

C-MATCHINOS 155
Graph C - {X, i!) corresponds to a bipartite graph H - {X, X, E) where
«y t ,5;] E if, and only if, {xlt x,) e V. The proof follows when Theorem 6
j'applied to the bipartite graph //.
Q.E.D.
Remark. Consider a l-graph G such that da(x) ■ dj(x) for each vertex
v. Such a graph defines a permutation of degree n, and vice versa. Thus.
Theorem 7 generalizes the well known theorem from algebra: Every permuta-
permutation is a product of transpositions.
3. Maximum cardinality of a c-matching
Consider a nuiltigraph G - (X. E) with a c-matching Eo <= E, and the
corresponding multigraph R(E0) with dark and light edges (see Section I).
Recall that the origin x0 is joined to some of the vertices of X by dark and
light lines, the edges of Eo are dark, nncl the edges of FQ - E — Eo arc light.
Consider the chains starting at the x0 and beginning with a dark edge.
II' there exists an alternating chain n starting at x0 and going to .ve X,
then orient each edge in ;< toward .v. li is possible that an edge will be given
two opposite directions. IT vertex x is the endpoint of a dark edge oriented
toward .v and is not the endpoint of any light edge oriented toward .v, then
vertex x is called dark. The set of all dark vertices is denoted by X".
If vertex .v is the endpoint of a light edge directed toward x and not the
endpoint of any dark edge directed toward .v, then vertex is called light.
The set of all light vertices is denoted by A"'.
If a vertex .v is the endpoint of a light edge directed toward x and also the
endpoint of a dark edge directed toward x, then x is called a mixed vertex.
The set of all mixed vertices is denoted by A"". Finally, if a vertex x is not the
endpoint of any edge directed toward .v, then vertex x is called inaccessible.
The set of .ill inaccessible vertices is denoted by A'1.
Vertex x0, the origin of the network R(E0). is so far unclassified: it will
be defined to be tight if there exist no alternating chains from x0 to xQ with
dark edges at each endpoint; otherwise. .v0 is defined to be mixed.
Eaeh vertex of K(E0) is either light, dark, mixed or inaccessible.
I hcoreni 8. Two dark vertices can onty be joined by a dark edge. Two light
v"rtices can only he joined by a light edg \ A mixed vertex and an inaccessible
fertex cannot be joined by an edge. A dark vertex and an inaccessible vertex
TO" Ol)ly be joined by a dark edge. A light vertex and an inaccessible vertex
co« only he joined by a light edge.
This follows immediately from the definitions.

I56 ORAPHS
These results are summarized below:
Type of vertex
Dark
Dark
Dark edge
Light
Mixed
Light
Mixed
Inaccessible
Dark edge
Light edge
Light edge
Inaccessibly
Dark
Light edge
Consider the subgraph of R(E0) generated by the set A"\ Denote the con-
connected components of this graph by A/,, M Chain ft is snid to enter A/,
via edge [a, h] if \i is of the form
H ■ [jfo.tfi.tf2 «* ■ o. bl - b,b2 b,]
with
■*o»  < . ■••' ak 4 '^i
bi, b2, b},.... /'(e M,,
I emma 1. Let xo$ A/, flm/.veA/1, If an alternating chain //[jco,.v] enter
Mi via an edge [a. b) and terminates with a dark (respectively, light) edge,
then there exists an alternating chain ft'[x0, x] entering A/, via [a, h] and
terminating with a light (respectir 'iy. dark) edge.
If a- is a mixed vertex, then there exists nn alternating chain v[.v0, x] ^
terminates with a light edge. Since x0 A/| and xeMi. there exist in
v[.t0. x] edges with one endpoint in X — A/, and the other cnclpninl in Mi
Let [r. </] be the last edge of this type in v[.v0, .vj. ■

C-MATCHINOS 157
Finally, let y be the first vertex of/* that is on v[d, x] (such a vertex always
exists)- If n[l>,y) and v[d,y] terminate with edges of the same type, the
theorem is true because of the alternating chain:
n' - nUo.y) + v[y,x).
\tn[b,y] a»d v[d>)'] terminate with edges of different types, then [a, b] -
[f>/) since, otherwise, tt[xo,y] + v[y,c] would be a simple alternating
chain, and c would be mixed, which contradicts c 0 A/,.
Thus, the chain
H' - n[x0, b] + v[d, x]
is alternating and satisfies the requirements of the theorem.
Q.E.D.
Lemma 2. Let xo$ A/,, xs Mi; let [a, b] be an edge incident to A/, and
directed into Mx. There exists an alternating chain n[xo,x] from x0 to x
that enters A/, via edge [a, h].
Let Y be the set of vertices of A/t that arc accessible by an alternating chain
entering via [a, b). Let Z be the set of other vertices of A/,. Since be Y, we
have Y jt 0. Suppose that Z »» 0, then there exists an edge [>\ z] with
ye Y and zeZ. We shall show thut this leads to a contradiction.
From Lemma I, y is accessible by two alternating chains /d[.v0,>'] and
HiUcn)'] entering A/i via edge [a, b) and terminating respectively with a
dark and light edge. Thus, z is accessible to an alternating chain entering
via («,/;]. This contradicts zeZ.
Q.F.D.
rheorcm 9. (Callai [1950]). / el i:0 l>e a set of dark edges in Gx. Let Mx be a
component of the subgraph of R( /;0) generated by the mixed vertices. lfx0 A-/,,
there exists exactly om edge incident to A/, and directed into Mx. Ifx0 e A/,.
'here exists no edge incident to A/, and directed into Mi.
'• If x0 ^ Mi, let ft be an alternating chain going from .v0 to Mi. Let b
to the first vertex of n in A/,. The alternating chain p[xo.b] enters W, via
an edge [a, b] that is incident to A/, and is directed into A/,.
Let [r, d] be an edge other than [a. b] that is also incident to Mi and
directed into A/,. From lemma 2. we know that de Mx is accessible by an
"Itcrnnting chain entering via [a, b]; from Lemmn 1, we may assume that
•"is chain terminates with an edge of a type different from [r, d]. Thus r
ls a mixed vertex, which contradicts c $ A/,.
Thus [a, b] is the only edge that is directed into Mt.

158 OKAPHS
2. If xoe Mx, then we can return to the preceding case simply by adding
vertices a0 and b0, a dark edge [ao.bo] and a light edge [bo,xo]. Let
replace x0 as the origin. The only edge entering A/, is [bo.xo). Thus there
exists no edge of the original graph that enters A/,.
Q.E.D.
Theorem 10. Let Eo be a maximum matching, and let Mi he a component
the subgraph generated by the mixed vertices in the multigraph R{L0). Tlwr
is a dark edge incident to A/, and directid exclusively into A/,. All oth r
edges incident to Mx are light and exclusively directed out of Mt.
1. Since the matching £0 is maximum, .v0 is light, and .v0 £ A/,. From
Theorem 9, there exists exactly one edge directed into A/,. Denote this edge
by [r, a], where c £ A/,, x e Mx.
Suppose [c, x] is light: since x is mixed, there exists an alternating chain
H from x0 to x that terminates with a dark edge. Chain n necessarily uses
edge [c. .v) (which is the only possible entrance edge to A/,).» followed by
dark edge incident to x and terminates with another dark edge incident to
But this is impossible, since there is only one dark edge incident to x. Thus,
edge [c, x] is dark.
2. Let [y,b] be nnothcr edge incident to A/| with ye Mi and b A/,.
From Theorem 9. [y, b) is necessarily directed out of A/,. Furthermore,
[y. b) cannot be dark because y, being a mixed vertex, can be reached by an
alternating chain terminating with a dark edge.
Q.E.D.
Corollary 1. l/E0 is a maximum matching, and if Mx is a component of the
subgraph generated by A"\ then | A/, | > 3 and \ Mi \ is odd.
Let [c,x] be a durk edge incident to A/, with .v A/,. Then. | .*/, | X
because, otherwise, .v cannot be reached by an iiltemating chain terminating
with a light edge. Furthermore, since A/, consists of vertex .v and pairs of
vertices joined by dark edges, | Mx \ is odd.
Q.E.D.
Corollary 2. If Eo is a maximum matching, each non-mixed vertex that
adjacent to a mixed vertex is a light vertex.
If [c.x] is a dark edge incident to a component A/, with c Mx, and
x e A/,, then c £ X1, c £ X" and c X". Consequently, c e A".
If [a, h] is a light edge incident to A/, with a $ Mi. b e A/,. then similarly,
a X',a<t X'1, and « $ Xm. Thus, a e A".
Q.E.D.

C-MATCIIINOS 159
Corollary 3. If Eo is a maximum matching, each vertex adjacent to a dark
vertex is light.
Let a e X". If [.v, a] is a dark edge, then xf X' (since x is the endpoint of
a directed edge); a* $ Xd (because, otherwise, a would be mixed), and x $ A
(from Corollary 2). Thus x e A".
If [.v, a] is a light edge, then x £ A", x i X", and x * A"* (from Corollary 2).
Thus.-teA".
Q.E.D.
Theorem 11. Let Eo be a maximum matching and let li be a c mnected com-
component of the subgraph generated by the inaccessible vertices. Each edge inci-
incident to lx is light and undirected. Each vertex x $ l^ that is adjacent to a vertex
d/7, u light. If graph G has no isolated vertices, then | /, | is even and > 2.
1. If [<\ x] is an edge incident to /| with xe /,. then c X", c X', and
c £ X" (from Corollary 3). Thus, c A", and [c. x) is light.
2. Since /| does not contain .v0 nor unsaturated vertices, it contains only
pairs of vertices joined by dark edges. Thus. | /, | is even und > 2.
Q.E.D.
Theorem 12. (Bcrgc. [1958]). Giren a simple connected graph G and a subset
S of the vertices, let p,(S) denote the number of components of odd onler in the
subgraph generati d by X — S. The number of unsaturatcd vertices in a maxi-
maximum matching is
{-max(p,(S)-|S|).
1. Consider a set S c A", and let C\. Ca..... Cp be the components of odd
order in the subgraph generated by X - S. If a component CH has no un-
unsaturatcd vertices, there is at least one dark edge going from Ck to a vertex
J* 6 5 because \ Ck\ is odd. Two distinct components Ck correspond to
two distinct vertices sk. Thus, n0. the number of unsaturated vertices, satis-
satisfies
pt(S) - n0 < (number of C* without unsaturatcd vertices) < | S |.
Hence,
\S\*h0 [ScX).
2. We shall show that 5 can be chosen so that p,(S) - S - n0 (this estab-
■ishes the theorem).
Let £0 be a maximum matching, and let X' be the set of light vertices.
prom Theorem 10 and Corollaries I. 2, and 3,
PAX1) - (number of components in Gx*) + \ X* \ .

160 ORAPHS
Furthermore, the dark edges of Ci define a bijection between the set
X' and the components Mk that contains no iinMitiinitcd vertices and no
dark saturated vertices. Thus
| X11 - (number of components in Gxn) + | X" \ - n0.
Hence
«o - pW) -\X'\.
Q.E.D.
Corollary 1. In a simple connected graph G - (X. E) of order n, the numbe
of edges in a maximum matching equals
where
- max |>,(S) - | S |
Self "
The proof is immediate from Theorem 12
Corollary 2. (Tuttc II947J), A necessary imd sufficient condition for a con'
nected graph to possiss a perfect matching is that
PAS)*\S\ (SeX).
The proof is immediate.
The following result generalizes the Pctcrscn thedrem [1891] for regular
graphs of degree 3.
Theorem 13. Let G - (X, E) be a connected multigraph that is regular of
degree h, with an even number of vertices, without loops, ami with
mG(S, X - S) > A - I (S c X: S * 0, A').
Then G possesses a perfect matching. Furthermore, eath edge of G is fr
(i.e. each edge ofG belongs to at least one perfect matching but does not belong
to all perfect matchings).
I. By using Corollary 2, we shall show that there exists a perfect matching-
I ct 5 be a non-empty subset of A\ where 5 X, and let d, Ca, ■■• •*
the connected components of odd order in the subgraph generated by X - &
By hypothesis, ;hoE, C\) > h - I. However,
«ioE. C,) > /) - I ,
because, otherwise, the number of edges in C» would equal
"UC.. Ct) - ^ [ft I C, | - (h - 1)] - -2 [/.(| C, | - 1) + 1]

C-MATCHINOS 161
which is not an integer, since | d | is odd. Thus,
mc(S. C,) > h .
Hence,
h\S\> »'c(S. X-S)> mu(S, CtvC2v •■■) - £ mu(S, Ck) > hp,(S).
Thus. p,(S) < I 5 | for all S X, S 0. This inequality remains valid
when S • X because p,(X) ■ 0, or when S ■ 0 because p,@) - 0 (since
the graph has an even number of vertices and is connected). Thus, from
Corollary 2. there exists a perfect matching £0.
2. To show that an edge e e E is free, it is sufficient from Theorem 4 to
show th.it e is on an alternating cycle relative to a perfect matching Eo.
We may assume that ee E - Eo (because if ee Eo and if some edge of
E - Eo that is adjacent to c is on an alternating cycle, then edge e will also
be contained in an alternating cycle).
We shall suppose that e is a light edge of C that docs not belong to any
alternating cycle, and we shall show that this leads to a contradiction.
Contract the endpoints of edge e into a single vertex x0 and define a trans-
transfer network R(E0) with origin x0. Since edge e appears in no alternating
cycle, vertex x0 is light.
Assume that there exist mixed vertices; let C\ be a connected com-
component of the subgraph of R(E0) generated by the mixed vertices. Clearly,
xt$ C,. Since | C\ | is odd from Corollary I to Theorem 10, we know from
Part 1 of this proof that
fnb(A'-Cl.Cl)>A.
Furthermore, from Theorem 10. there is a single dark edge of R(E0) thai
leaves C,, and thus there arc at least h - I light edges that leave C\.
Now consider the network R(E0) obtained from R(E0) by contracting each
component Ck of mixed vertices. Thus, Ci becomes a dark vertex c,, and
no edge has been oriented in two directions. From Theorem 10, r, is incident
to a dark edge directed into r, and to light edges each directed out of cv.
Let X* and J" respectively denote the sets of dark vertices and light vcr-
•jees in network #(£„). Let d,*(x) denote the number of light edges of R(E0)
Greeted out of .v. Consequently.
dj(x) > h - I CveX").
Asides, the number </f(.v) of light edges directed into x satisfies
dj(x) < </«(*) - 1 - h - 1 (x e X' - { x0 }).

162 GRAPHS
Thus
15*1 (fc - ix I </;(*■) - £ </,-(*) < (/• - i) | x' - {x0} I + w,-(x
<(/»- I) I A" I -(/»- I) + 2(/»-2).
Since the theorem is obvious for h - I. we may assume that h > I; thu
\X'\<\X'\ + I -^-ry-
By counting in two diflcrcnt ways the number of dark edges of /?( 0)
that have exactly one direction, we obtain
|X('|-|X'I+ I ■
Since we have obtained two incompatible relations, the proof is achieved.
Q.E-D.
Corollary (Errcra [1922]). If G — (X, E)ls i simple conneclnl graph regular
of degree J such that all the isthmi of G are on the same elementary chain,
then G possesses a perfect matching.
Recall that an "isthmus" is an edge [a, h] whose removal disconnects the
graph. In a connected graph, the removal of an isthmus creates exactly two
connected components.
Since 3 | X \ m 2 | E |. the number of vertices of G is even. Consider the
different connected components created by the removal of all of the isthmi.
From the hypothesis, each of these connected components is joined to the
rest of the graph G by one or two isthmi.
Suppose a connected component C, is joined to the rest of the graph by
two isthmi [x,c] and [y,c'\, where c.c'eC, and x.y Ct. Consider the
graph 6', obtained from graph Gc, by joining vertices c and c\ Graph i
is connected, regular of degree 3 and has no isthmi. Therefore, from Theorem
13. graph G, possesses a perfect matching that contains edge [c.c'\. Thit
matching corresponds to a matching £0(Q of graph C -, that saturates all
vertices except c and <•'.
Suppose a connected component I), is joined to the rest of the graph by
only one edge [</. .v] with </e D,. xi Dr Consider the graph //, obtained
from GD( by removing vertex tl and by joining the vertices <l, and d-i of
that arc adjacent to vertex <l in graph G. l-r'om Theorem 13, graph h', hoi
a perfect matching that docs not contain edge [c/,. <l3], and that correspond!
in graph GDf to a matching E0(D,) that saturates every vertex except tie Df
Thus, a perfect matching for G can be formed from the union of sett
U £o(C). U E<JLD,). and the set of isthmi or graph G.

C-MATCHINOS 163
EXERCISE
1 Show the following result:
Given a multigraph G - (X, £). there exists a partial multigraph H with rfM(x,) -
Ml) if, and only if. for each pair 5, T of disjoint sets, the number q{S. T) of components
C of the subgraph of C generated by X - S - Twith mo(C, T) + rf(C) odd satisfies
q(S, T)
Apply Theorem 5 to the graph G constructed from C as shown in Theorem I.
(W. T. Tutlc |1954])

CHAPTER 9
Connectivity
I. /i-Coiinccted graphs
The connectivity k(G) of a connected graph G is defined to be the minimum
number of vertices whose rcmovul disconnects G or reduces 6' to a single
vertex.
If G is not a clique, there exist two non-adjacent vertices a and b; thus,
X - [ a, b} is a set whose removal disconnects G, and, consequently,
k(G) *\X- {a,b)\mn-2.
On the other hand, if G is the /i-clique Kn, we have
K(Kn) - n - I .
Graph G is said to be h-connected if its connectivity x(G) is > h. An
articulation set of G is any set of vertices of G whose removal disconnects G.
Let denote the family of all articulation sets of G. A graph G Kn\% •
connected if, and only if,
k(G) - min | A | > h .
A4J/
Thus G is h-connected If, and only If,
A) h < n - I, and
B) //kw « «o articulation set AofG with \ A \ - h - I.
Theorem 1. for a simple connected graph G,
k(G) < min do(x).
If G has « vertices, then k(G) < « - I. Let x0 be a vertex of minimum
degree h, and suppose thut h < k(C). The set ro(xo) has cardinality h < n 1,<
Thus G is not a clique, and
- min | A \ < | rc(x0) | - ii
164

CONNECTIVITY 163
hich is a contradiction.
* Q.E.D.
Ihcorcm 2 (Harary [1962]). For each m, n with 0 < n - I < m < (I,
.kg maximum connectivity for a simple connected graph with n vertices and
r 2 m l
„ edges is [ n-j.
I, From Theorem I, a connected graph C satisfies
2 m ■ £ dc(x) > n min </<,(.*) > h / k(G) .
x« X x« X
Hence.
and so
2. Note thut, if C is a simple connected graph, then
n- 1 <m
It now remains to show that for each n, wi, satisfying these inequalities there
exists a simple connected graph G(n, in) such thut
K(G(n,m))-[2™]
Case I. Suppose — is un integer. Consider two subcases:
n
Subcase I'. Suppose — ■ 2k is an even integer. Construct a graph
°».a* with vertices x0, x, *»-i. Join each vertex x, to the vertices of
{xjljm i± k' (mod. n) ; I < k' < k }.
met euch vertex has degree 2 k, the number of edges in graph Gn,M is
m{Gm,n) - j I dCm Jxt) m kn - m .

166 OKAPHS
It can be easily shown that this graph is B Ar)-connectcd.A) Thus, take
Subcase I*. Suppose that — - 2 k + I is an odd integer. The
n
mB k + I) - 2 m implies that n is even. Form graph G(n, m) by taking grap
Gn,m and by joining vertex x, to vertex x, if
J m i + -^ (mod. n),
y-o,i !■- '•
Thus, for G - G(n, m),
m(G) - ^ X <M*i> - \ B k + I);. - m .
It can be easily shown that this graph is B At + l)-connccted.'a>
^ .» „ , 2 »i. . , r 2 m i
Cask 2. Suppose that — is not an integer. Let q - — .
n I n i
Subcase 2'. Suppose that n or ^ is an even number. Form graph G(n, m)
by taking graph Cn., and adding m - nq edges arbitrarily.
<" We shall show ihat Gn,,k is B fc)-connccicd. From ihe lymmeiry of G, it is sufllclcni to »how
thalxa*ndxala » 1,2 n - 1,cannoibc disconnected by leu lhan 2 A vertKci. SupposeiliM
x0 and x can be disconnected by 2 ft - 1 vertices. xt,, x,a, ...mx,ak.l where
I < »i < »i< ■■•< /i*-i < n— I.
One of ihe two intervuli @, a], (a, /?| contains ai moil k - 1 oflhoe indicei. Suppose ii is inierv
@, a|. Then, two consecutive vertices of ihe sequence obtained by removing *,,. x,a... from Xn
x x,, are Joined by an edge (because the difference between (heir indicei ii < k — 1). Con
qucntly. there ii ■ chain from x0 to xu, which ii a contradiction.
'■' We shall show ihat CBlW+1 isBfc + 1) connected. Suppose that xa and x. are disconnected
by 2 k vertices x^.Xi, *,,„ where 1 < /i < /t < ... Igi, < n — 1. lfoneof the two inlervi
[0, a] (a. n| does not contain k consecutive indices, then a chain from x0 to xa can be conilruclrf
as shown above. Therefore, suppose that xn und xa are disconnected by x,, x,tl
where 1 < / < a - ft + I. and
Xutjs X«|/«.|. ...a Xfl4^4 M- I •
where
1 <;< n — k—9.
Let
[ y] /?'/?+; (indices mod. id.
Then,
; + *.« + /—I). /!'«!/+ «(,«+;— 1|.
There is a chum from xo to x, and from x,- lo x.. Hence, there is a chain from x0 to x. sla
lx,, xf) is an edge of Cn, lt.,.

CONNECTIVITY 167
Subcase 2". Suppose that both n and q are odd integers. Let q - 2 k + 1.
form the graph G(n, m) by adding to Gn,m n edges or the form [xt, xj\, where
. M / h -— modulo n, and also /« - nq arbitrary edges. The total number
of edges now equals (m — nq) + n + 2kn - m. Similarly, it can be shown
that this graph G(n, m) is ^-connected.
Q.E.D.
The following theorem for /i-conncctcd graphs was discovered first by
IC. Monger [1926] and rediscovered by H. Whitney [1932].
L«mma If a and b are two non-adjacent vertices of a simple graph G, then
ihe maximum number of vertex-disjoint chains between a ami h equals
where the minimization is taken over all vertex sets A that do not contain either
a orb and whose removal disconnects a from b.
To prove the lemma, we shall construct a transportation network R with
source a and sink b in which the flow tp that maximizes </>, determines </>,
vertex-disjoint chains from a to b in G.
First, replace each edge in G by two oppositely directed arcs with the same
endpoints. Replace each vertex x a.bby two vertices x' and .v" joined by an
arc (x',x") with capacity I. Replace each arc entering x by an arc enter-
entering x' with capacity I, and replace each arc leaving .v by an arc leaving x" with
capacity I.
Thus, the maximum flow between a and b represents the maximum number
of vertex-disjoint paths between a and b. The minimum capacity of a cut
between a and b represents the minimum number of vertices needed to dis-
disconnect a from h. From Theorem (I, Ch. 5), these quantities are equal.
Q.E.D.
'"hcorcm 3 (Mcngcr [1926]). A necessary and sufficient condition for a
simpie graph to be h-connected is that any two distinct vertices a and b can be
joined by h vertex-disjoint chains.
Let A be a positive integer. Since the proof is obvious for h - 1. suppose
* > 2.
Necessity. Let G - (X, E) be a //-connected graph. The lemma shows that
* non-adjacent vertices can be joined by h vertex-disjoint chains. We shall
that this is also true for two adjacent vertices.
Suppose this were not true. Let a and h be two adjacent vertices that cannot

168 ORAPKS
be joined by A vertex disjoint chains. Consider the graph C obtained fro
by removing edge [a, b]. Vertices a and b cannot be joined in C by A .
vertex-disjoint chains. Thus, from the lemma, there exists in C a ,
A c X - {a, b} of cardinality < A - 2 that disconnects a and b. Then
| X - A | - | X\ - | A | > (A + 1) - (A - 2) - 3,
and there is a vertex c distinct from a and b in X - A.
We shall show that it is possible to join vertices c and a in graph C by
chain that avoids A. If c and a arc adjacent, clearly this is possible. If c and
arc not adjacent, they arc joined by h disjoint chains in G, and therefore, th<
are joined by A - 1 disjoint chains in G'\ one of these chains avoids \
because | A | < h - 1 .
Similarly, vertices c and b can be joined by a chain that avoids A. Finally
vertices a and b in C can be joined by a chain that avoids A, which contra*
diets the definition of A.
Sufficiency. If two arbitrary vertices are joined by h disjoint cnains, graph
is connected. At most one of these chains has length 1, and the union oft1,
vertices of the A - 1 other chains contains at least A - 1 distinct vcrti
other than a and b. Hence,
n >(/i - 1) + 2 > h.
Thus, h < n - 1.
Furthermore. G has no articulation set A with | A \ < h because, oth
wise, we could choose two vertices a and b in two distinct components oftr
subgraph generated by X — A; each chain joining a and b passes through set
A, but there cannot be A distinct vertices in A.
Thus G is A-connected. n £ rj
Corollary I. ifGis h-connecled, the partial graph obtained by removing 1
is (A - l)-connected.
If two vertices can be joined by A vertex-disjoint chains in G, they can be
joined by at least A - I vertex-disjoint chains in the new graph.
Q.E.D.
Corollary 2. l/G is h-connected, the subgraph obtained by removing a v rteX
is(h — l)-conncctcd.
If two vertices of the new graph can be joined by A vertex-disjoint chains W
G, then they can be joined by at least A - 1 vertex-disjoint chains in the
graph.
Q.E.D-

CONNECTIVITY 169
Corollary 3. Let Gbeasimpleh-connceted graph. Let B - {bx,b , bh}
to a &t of vertices with \ B | - A. If a e X - B, there exist A rertex-disjoint
tlementary chains n\a, b,) joining a and B.
We may assume that none of the b, arc adjacent to a because, if k vertices
of B arc adjacent to a. then h can be replaced by h — k in the theorem.
Let graph C be the graph obtained from graph G by adding a vertex z and
joining it to each vertex A, 6 B. We shall show that graph G' is A-connected.
Let 5 be a subset of X with \S\ < h — 1; we have then to show that the sub-
subgraph generated by X u {z} - S is connected. This is clearly true if z 6 S,
since Gx.s is connected. If z $ S. then, since \S\ < A, one of the b, docs not
belong to St and since this bt is adjacent to z, the subgraph under considera-
consideration is connected.
From the lemma, there exist, then, h vertex-disjoint chains in G' between a
and z. These chains induce in G the required chains n,[a, b,].
Q.E.D.
Corollary 4. Let G be a simple h-connectedgraph with h > 2. An elementary
cycle passes through an arbitrary set of two edges ex and ea and h - 2 vertices
1. If A — 2, the graph G' obtained from G by adding vertices a and b in
the middle of edges ex and ea. respectively, is again 2-connected because no
vertex of C can disconnect C. l-'rom the Mcnger theorem, there exist two
vertex-disjoint chains in G' that join a and b. These chains determine an
elementary cycle passing through edges e, und e3t and (he proof is achieved.
2. Suppose // > 2 and suppose that the theorem is true for all ^-connected
graphs with k < h. We shall show that it is also (rue for a given h-
«onnccted graph G. We may assume that au oa, ...,fli,.a are not the end-
poinis of either e, or c2 since then the theorem would follow from the induc-
induction hypothesis.
The subgraph obtained from G by removing vertex aH.a is (A - 1)-
connectcd. from Corollary 2. Hence by the induction hypothesis, there is a
Cvc'e /<o passing through r,, e2, ax, oa,..., a*-a. Let B be the vertex set of
cttjcna. Clearly, \D\ > A.
From Corollary 3, vertex a,,_a and set J?can be joined by h vertex-disjoint
'tains. Suppose that each of these chains encounters B at only one vertex
0-c. does not return to B). Denote these chains by

170 ORAPHS
Suppose, Tor example, that cycle /i0 encounters vertices
t>i, bit •■■! bi,€ B
in this order. Among the h segments of /i0 determined by two consecuti „
vertices of the sequence bx,b , bh, blt there is at least one segment that
does not contain in its interior any of the h — 1 elements a,, a ^-s.^.e,,
For example, let /io[6i. ba) be this segment. The cycle required for the proof
is then:
QE.D.
Corollary 5 (Dirac [I960]). If G is a simple h-connectedgraph with h
then there is an elementary cycle passing through h arbitrary vertices.
Select any set of h vertices au aa,.... ah. Consider an edge [ah.u x) and an
edge [ah,y]. From Corollary 4, there exists an elementary cycle passing
through these two edges and through vertices alta , o».a.
Therefore, there exists an elementary cycle containing vertices ax, aa ak.
Q.E.D.
Theorem. (Halin [1969]). If G is a simple h-connected graph, either the
exists an edge such that the partial graph obtained by removing this edge is also
h-connected or there exists a vertex of degree h.
The proof, which is omitted, uses Theorem 3. For the cases h - 3 and ,
the theorem was first proved by Las Vcrgnas [1968].
We shall now study conditions on the degrees of a graph that imply
//-connectivity. The following result is a slight generalization of a result of
Bondy [1969].
Theorem 4. Let G be a simple graph with n vertices xlt x3, ...,xn such that
dcM < d^Xl) < - < dc(xK) - d.
Let /, < i3 < ••• be the sequence of indices i such that d^x,) < i. For the /-'A
index of this sequence, let
i
Then, the number p of connected components of G satisfies
- d
H
Let c, < ca < ■• < c, be the cardinalities of the vertex sets of the p con*

CONNECTIVITY 171
cted components of G. We may assume that p > I because, if p < /, the
DrOpos«l inequality is obvious.
Graph Ceontains at least ct vertices of degree < c, - 1, and ct + ca +
...+ a, vertices of degree < ck - 1. Hence,
do(x«., ♦«+•■■+*> < ck - 1 < c, + c2 + - + ck (fc - 1, 2 p) .
For k - l,2,...,/7, the numbers fi + ra+"•+<"* arc all different.
Hence, c, + ca +-+ ck > tk, and so
I ck > da(xt,+ti+....+ek) + 1 > <fc(*J +1 (fc - 1,2 p - 1).
[ c, > d + 1 ,
Since lip- 1. we have
' p-i
n - c, + c2 + - + cp > J do(jfll() + £ ^(x,,,) + d + p.
Thus, from the definition of q(l),
I do(xlh)>lq0)<
Furlhcrmore, k > l,v/c have </«,(*,„) > ^(/). Hence,
«> M0 + (p- 1 -/)<?(/) +d + p,
and thus
r n + g(l) - d
which implies that
Q.E.D.
Example. Consider the graph G in Fig. 9.1 with the following degree
sequence; 1, 1, 3, 3, 4, 4, 4, 4,4. The vertices x, with </<,(*,) < / are circled.
For/- 1, wehave<?(l) - I, and
For / - 2, we have ^B) - 2, and

172
ORAPHS
Thus, Theorem 4 shows that G cannot have more than 2 components.
v. x*
Fig. 9.1
Corollary I (Bondy [1969]). Let G be a simple graph with n vertic
xitXi, .,.,*„ such that
dc(x\) ^ da(xi) < ■" < da(xid ™ d*
If, for some integer q,
</o(*») > * (* - 1, 2, .,., (/),
/ton //»e number p of connected components of G satisfies
n + q - d
There exists an index / with do(xt) < i because dc(xn) < ;i; let /, be the
smallest such index. We have
' dc(xh) > dc(xq) > q.
Hence
a + ?A) - d n + q - d
q(l) +1 <7 + 1
The proof follows by letting / - 1 in Theorem 4.
Q.E.D.
The graph in Fig. 9.1 shows that Corollary 1 is weaker than Theorem •
Corollary 2. Let G he a simple graph with n vertices jc,. xa xn such thai
do(Xi) < <lc(*i) < ■' < rfo(*ii) " </•
> k far all k < ;/ - rf - 7, //kvj C fa connected.

CONNECTIVITY 173
in Corollary Met
q - n - d — 1 .
VVe have
n + q - d - (? + d + 1) + q - d < 2(q + 1),
and. from Corollary 1,
Thus graph G is connected.
Q.E.D.
Corollary 3 (Bondy [1969]). Let G be a simple graph with n vertices *,,
x xn such that
If for some integer h < n
dc(xk) >k + h-l (k*n- dc(xK.h+,) - 1)
then G is h-connected.
Consider a set A c X with cardinality h — 1. We shall show that A cannot
be an articulation set. Index the vertices x[ of C - GX.A so that
If* <\X \- d' - 1. then
* < (» - A + 1) - (to.ul) - /, + 1) - 1 - /. - dG(xK.h+x) - 1 .
l:rom the hypothesis, this implies that
««**) > * + * - 1 .
which implies that
dc(x'i) > rffif**) -|i4|>fe + fc-l-(fc-l)-fe.
Corollury 2 shows that Cx-4 >s connected. Since this is proved Tor each set
A c X with cardinality // — 1, graph G is /i-connectcd.
Q.E.D.
Corollary 4 (Chnrtrand, Kapoor, Kronk [1968]). Let G he a simple graph
°f order n such that, for some integer h < n, the following tn-c conditions hold:
(•) for each k < ^-y-' , the number of vertices with degree < k + h - I
is fas thank;

174 ORAFH3
B) the number of vertices with degree < —= I is less than n - h
Then graph G is h-connected.
After indexing the vertices as in Corollary 3, conditions A) and B) becoitfe
equivalent to
(lf)
The inequalities k > —s— and k < n - </e(Xn-* + i) - • cannot
satisfied simultaneously because this would imply that
n - h
and hence
n + h I ^ j/ \ n — h n + h
which is a contradiction.
Tl
0').
Thus, if k < // - dJLxn.h^l) — 1, then A: < ——— and, from conditioi
//0(jf,) > * + A - 1 .
Corollary 3 shows that graph G is /i-connectcd.
Q.E.D.
Corollary 5 (Chartrand, Harary [1968]). Let G be a simple graph oforderH
such that, for some positive Integer h < n.
Then G is h-connected.
To prove the corollary, it is sufficient to show that conditions (I1) and B1)
of Corollary 4 are satisfied. If k < ^-y—' . then, for each vertex x of grnph 6

CONNECTIVITY 175
Thus condition (!') is satisfied; condition B') is obviously satisfied.
1 Q.E.D.
2. Articulation vertices and blocks
A vertex whose removal from the graph increases the number of connected
components is called an articulation vertex. An edge whose removal from the
graph increases the number of connected components of the graph is called an
Isthmus. Using these definitions, we can redefine 2-conncctivity: A graph is 2-
coniKctctt if, and only if, it is of order n > 3, connected, and has no articulation
i-eriices.
A set A of vertices in graph G that generates a subgraph CA that is con-
connected and without articulation vertices and is maximal with respect to this
Flu. 9.2. Cactui
Property is called a block. Thus, subgraph GA is cither 2-connected (if
M | > 2), or an isthmus of G (if | A | - 2). or an isolated vertex of G
(if | A | - 1). it is left to the reader to verify that the graph in Fig, 9.2 has 9
"fticulation vertices, 6 isthmi, and 13 blocks.
A chord of an elementary cycle is defined to be an edge that joins two non-
consecutivc vertices of the cycle. A cycle of length 3 has no chords. A cycle
°f length 4 can have 0, 1 or 2 chords.
. A cactus is defined to be a connected graph in which every block is either an
■Mhrnus or an elementary cycle without chords (sec Fig. 9.2).
The principal characteristics of 2-connected graphs arc described in the
"ollowing theorem.

176 ORAPHS
Theorem 5. The following properties are equivalent in a graph G of r
>3:
A) G is 2-connccted.
B) Every two vertices ofG lie on a common elementary cycle.
C) Every vertex and edge of G lie on a common elementary cycle,
D) Etwy two edges of G lie on a common el mentary cycle.
E) Every two edges ofG lie on a common elementary cocycle.
F) Every two adjacent edges of G lie on a common elementary cycle.
A) => D) This was established by Corollary 4 to Theorem 3.
D) » C) B) This proof is obvious.
B) A) This follows from Theorem 3.
D) (S) Let [a. b] and [r. d] be two edges, and let
[b, a, a,,a2 ak, c, d, dt,d2 d,, b]
be an elementary cycle that contains them.
Let A - { a, ax aK,c}. Consider the subgraph GX.A, and let fldeno
the connected component of GX.A that contains b and d. Then cocycleuK
in G is an elementary cocycle that contains edges [a, b] and [r, d).
E) => F) Let [a, b] and [a. c) be two adjacent edges. Let m{A) be an clcmem
tary cocycle that contains these edges and that separates the graph
into two connected components A and B. If ae A, then h Band
ce B. Consequently, there exists an elementary chain /<[£>, r] it)
B that joins b and r. I lencc,
n[b,c) + [c, a) + [a,b]
is an elementary cycle that contains edges [a, b] and [a, c).
F) (I) We shall show that G is 2-connectcd. Otherwise, G has an articula-
articulation vertex a. Let B and C denote the two connected components of
the subgraph Gx _,„,. Vertex a is joined to B by an edge [a, b) and td
C by an edge [a, c); clearly, no elementary cycle can contain both
these edges. This contradicts F).
Q.E.D.
In the following theorem Ramachandra Rao has chdractcrized the simpk
connected graphs with n vertices and r articulation vertices that have the
maximum number of edges.
Lemma. A connected graph of order n > 2 has at least two vertices that a
not articulation vertices. There are exactly two such vertices if, and only
the graph consists of an elementary chain.

C0NNKCT1VITY 177
prom Chapter 3, Theorem 2, we know that a spanning tree of G has at
least two pendant vertices. Clearly, these vertices cannot be articulation
vertices of G.
\\G has only two vertices that arc not articulation vertices, then each span-
spanning tree of G has exactly two pendant vertices. This implies that G is an
elementary chain.
Q.E.D.
Theorem 6 (Ramuchundra Rao [1968]). In a simple connected graph of
order n > 2 with r articulation vertices, the maximum number of possible edges
is
CD
Recall from the lemma that n - r > 2; thus this formula is meaningful.
If C is such a graph with the maximum number of possible edges, then
each block of G is a clique with at least two vertices. If there arc/? blocks, then
p > r + I. Let n, > 2 be the number of vertices in the Mh block. We see
easily, by induction on the number of blocks, that
£ «, - n + p - 1 .
Thus, the number of edges of G is
max ( X (?) / I "i - " + P ~ I = »i. "j », > 2; p > r + 1 !
('
■«('—
-CD*'-
Note that we can construct a graph with r articulation vertices and with the
Maximum number of possible edges by taking a clique Kn.r with n - r
^wtices and attaching to one of these vertices an elementary chain of length r.
"us, there arc r articulation vertices, and the number of edges is
Q.E.D.

178 UKAPHS
We shall now apply the preceding results to graphs in which each ele t>
tary cycle of even length has at least two chords. Trees and cacti with only
cycles arc examples of such graphs. Other less trivial examples will be enco
tercd later.
Lemma 1. If each eivn elementary cycle has at least two chords, then a 4
even elementary cycle generates a clique.
If this were not true, then there would exist an even cycle n or minimum
length that does not generate a clique. Let /1 - [at, aa ap, a,].
Clearly, p > 6, since Tor p - 4, a quadrilateral with two diagonals '
clique.
An even chord (respectively, odd chord) is defined to be a chord that dividet
H into two even chains (respectively, odd chains). There arc only two cases
consider:
(I) Cycle ft has an odd chord [a,, a,]. Then the cycles
arc even and or length less than /(/;), the length of/a. These cycles generate t
cliques (see Fig. 9.3).
Vertices
fl;€/i[fl,,fl,_,] and flte/i[a, + ,,flp]
are adjacent because the cycle
[a,, aj) + [a}, a,] + [a,, ak] + [ak, at)

CONNECTIVITY
179
quadrilutcral and therefore possesses two chords, one of which is ncccs-
[a,, flj- Thus n generates a clique.
B) Cycle n has two non-adjacent even chords [at. at] and [a,, ak]. Suppose
that these chords cross one another; for example, suppose
as in F'8- ^-4- Consider the following two even cycles:
r, - /i[fl,, aj) + [o;, ak) - n[a,, ak) + [«„«,].
v, - ft[aj. a,] + [a,, at] - n[ak. o,] + [a,. as)
a
Fig. 9.4
At least one of the cycles v, has length /(v,) with
I
9.5
/(»',) <
+ 2 .
«nce, otherwise,
Up) - /(v.) + /(v2) - 4 > j /(/i) + 2 +1
por example, let v, be this cycle v,. Then,
+ 2 - 4 - /(/<)
Consequently, vx generates a clique, and there exists in n at Icasl^ne odd
c"ord. From Part A) of the proof, n generates a clique.
" the two chords do not cross one another, suppose, for example,
" k. o,], and a, e [at, o,], where k < j (see Fig. 9.5). Consider the cycle
v - [a,. a,) + n[a,, ak) + [ak, at\ + n[aj, a,].

180 GRAPHS
Since cycle v is shorter than cycle /;, it generates a clique, and cycle
contains an odd chord. From part (I) or the proof, n generates a clique.
Hence, in all cases, /; generates a clique. This contradicts the definition of
QE.D.
Lemma 2. I/each even cycle has at least two chords, then an odd elementary
cycle with at least one chord generates a clique.
It is sufficient to show that if /j - [a,, a ap, at] is an odd cycle and'
[ai, a,] is one of its chords, then [o,, a{ _,] and [ot, a, +,] are also chords. It js
evident that one of the cycles
/(, - /i[fl,,fl,] + [a,, ax) or /;, - [a,,a,] + nla,,at]
is even. Suppose /ij is even. From Lemma I, we know that Hi generates a
clique and this implies that [fl|.fl,_i] £. Since [at,«._,]e E, consider the
cycle
H' - [01,0,-1] + n[a,.i,a,].
This cycle is even, and so we know from Lemma I that [«,, a, „,] e £.
Q.E.D.
Theorem 7. (Dirac [I960]). 1/each even elementary cycle has at least two
chords, then each block is either a clique or an odd cycle without chords.
Let B be a block of G. If B is not a clique, then there exist two non-adjacent
vertices A, and ba in B. Since GB is 2-conncctcd. there is a cycle /i0 that contains
bt and ba. This cycle /;0 is odd and has no chords (since, otherwise, 6t and ftj
would be adjacent, from Lemmas I and 2).
Let BQ c B denote the set of vertices on /;0- Suppose first that Bo B.
Let o e B - Bo. From Corollary 3 to Theorem 3. there are two vertex-dis-
vertex-disjoint elementary chains /(Jo, .v] and /ja[a, y] joining a and Bo, that contain no
vertices of Bo except x and y.
One of the cycles
/»i[«. x] + »0[x,y) - »2[a,y\
, x] -
is even. Since this cycle generates a clique (from Lemma I), vertices x and J1
arc adjacent. Since /j0 has no chords, x and y are consecutive vertices oftne
cycle. Therefore, the even cycle contains bt and b2. Thus ftt and ba are «""
jaccnt, which is impossible.
Thus Bo - B is an odd cycle without chords.
Q.E.D.

CONNECTIVITY 181
j, it-Edgc-conncctcd graphs
In this section we shall study, for a gcncrul graph G - (X, U), the car-
cardinality or a minimum cut between two vertices a and b, i.e. the number:
, b) - min »icM, X - A).
aoA
btX-A
e (-4, B) denotes the number of arcs going from A to B. We shall also
study, for » multigraph G - (A', £), the number:
cv(a, b) m min mc(A, X - A).
AcX
»*A
t>*X-A
I Iworcm 8. I. Let G - (X, V)bea graph; then cj {a, b) equals the maximum
number of arc-disjoint paths from a to b.
2. Let G — (X, E) be a multigraph: then ca(a, b) equals the maximum
number of edge-disjoint chains from a to b,
1. If G ■ (X. U) is a graph, form from G a transportation network R with
a as a source a and b as a sink. Let each arc tie V have capacity <-(») - I,
and add a return arc (b, a) - up with infinite capacity.
In network R. the maximum flow problem (Ch. 5. § I) is to find a flow tp,
satisfying 0 < <p{u) < 1 for all u in V, that maximizes the flow value in the
return arc w0-
Theorem (I, Ch. 5) shows that
max <p(u0) - min m£(A,X - A).
9 Ac X
Ama
At*
The left side of this equality equals the maximum number of arc-disjoint
Paths from a to b in graph G. The right side is. by definition, cS(a, b). This
proves Part 1 of the theorem.
2. If G - (A', £) is a multigraph, consider the graph G* obtained from G
°y replacing each edge of G by two oppositely directed arcs. The maximum
"umber of elementary chains in G between a and b with no common edge
t(luals the maximum number of elementary paths in G* from a to b with no
c°mmon arc. From part I, this number equals
, b) -
Q.E.D.

182 ORAPHS
Nash-Williams I cmma. IfG - (X, E) is a multigraph, it is always
to construct a set E' of new edges that matches the vertices of odd degree, and
such that graph C - (X, £') satisfies
min(mcM.X - A) - mo(A,X - A)) - 2[icc(fl, fc)] ,
n ■ il
Atb
for all as X, be X.
Let Y denote the set of vertices of odd degree, and let Z denote the set <
vertices of even degree; then
2 | £ | - I dc{x) - I </0(x) + £ dG{x) .
xTX icV xtZ
1 hus | K | is an even number, and it is always possible to match together tht
vertices of Y. The proof of this lemma is long, and the reader is referred j
the original presentation (Nash-Williams [I960]).
A connected multigraph G is said to be k-edge-connected if it cannot bcdi$.
connected by the removal of less than A; edges, i.e., if and only if
Co(x, y) > k for all x, y e X, x y.
A multigraph is I-edge-connected if, and only if, it is connected. A multi
graph is 2-cdgc-conncctcU if, and only if, it is connected und has no isthmus.
The least k such that C is ^-edge-connected is culled the edge-connectivity.
I cmma. In a connected multigraph C, an edge is an isthmus if, and only ,
no elementary cycle of C contains this edge.
If [a, h] is not an isthmus, the graph remains connected after the removal of
[a, h] and there is an elementary chain f:[a, b] of C that does not contain
[a, b]. Thus, [a, h] together with //[a, b] form the required elementary cycle.
Conversely, if edge [a. b] lies on a cycle, then the removal of [a, b] does not!
disconnect the graph, and so [a, b] cannot be an isthmus.
Q.E.D.
Theorem 9. A connected multigraph is 2-edge-connected if, and only <
every edge lies on a cycle.
The proof follows from the lemma.
Theorem 10 (Robbins [1941]). Given a simple graph G, its edges can
directed to form a strongly connected I-graph H if, and only If, G is 2- "8 '
connected.
I. If such a graph //exists, then G is 2-cdge-connectcd because the remov •
of an edge cannot disconnect H and, consequently, cannot disconnect G.

CONNECTIVITY 183
2. Conversely, if G is 2-cdgc-connected, each edge of C lies on an elemen-
(ary cycle. Let Ax denote the set of vertices on a first elementary cycle of C.
Direct the edges of this cycle to form a circuit, and arbitrarily direct all the
edges with both endpoints in Ax. The subgraph generated by At is strongly
connected. If X - Ax, the proof is achieved.
If X - At »* 0, there is a vertex aa$ Av that is adjacent to a vertex
fl] A\ (because the graph is connected). By hypothesis, edge [fl>, aa) is on an
elementary cycle that contains a chain leaving A} at vertex ax and returning
t0 Ai at some vertex A,, no edge of which is already directed. Direct this chain
so that it becomes a path. Let Aa denote set Ax augmented by the vertices of
this path. Arbitrarily direct all undirected edges with both endpoints in Aa.
The subgraph generated by Aa is strongly connected. If X Aa, this
procedure can be repeated, as many times as needed, to obtain a strongly
connected graph //.
Q.E.D.
Applications of this result to traffic problems arc easily seen. If the
streets of a city are represented by a graph, then the streets can be directed
for one-way traffic without cutting off traffic between any two points, if,
and only if. the graph is 2-edge-conncctcd.
The above theorem can be generalized:
Theorem II (Nash-Williams[l960]). The edges of a multi-graph G - (X, E)
van be Urected to form a graph H - (A", U) with
Cu(x. y) >[2 cc(x, >■)] (x.yeX; x + y).
We may assume that multigraph G is connected; otherwise, each connected
component could be considered separately.
Case I. Each vertex of G has even degree. In this case, it is easy to see that
there exists a cycle /< that uses each edge exactly once. (This result, due to
Eulcr, is one of the oldest results in graph theory: a proof is presented in
Chapter II. § I.)
If the edges are directed along the direction of travel in /<. then a graph H is
obtained. For each A <= X. A 0. A X. we have
m*,(A,X - /O- mf,(A,X - A)
"Ccausc a traveller through ft exits set A as many limes as he enters set A.
Thus,
mc(A, X - A) - wijM. X - A) + m^(A. X - A)
X - A) .

184 GRAPHS
Consequently,
c*i(x, y) ■ min mf,(A,X - A) ■ - min mc(A, X - A) - »cc(*. >")
Atx *■ Ant *
which establishes the theorem for this case.
Case 2. There exist vertices of odd degree. The proof for this case dependg
upon the Nash-Williams Lemma given above. Add to C a set £' of edges
that matches the vertices of odd degree, as in the lemma.
Let C + C - (X. E u £'). Since each vertex of graph C + C has evj i
degree, its edges can be directed as shown in Case 1 to form a graph H +
IV - U,[/ut/') where // - (X, V) is graph G - {X, E) with its ed
directed, and //' - (X, V) is graph C - (X, £') with its edges directed. If 4
is a set of vertices corresponding to a minimum cut in // between two given
vertices a and b, then
cfa. b) - mfrA, X - A) - »C+H-M. X - A) - mfAA.X - A).
From Case I and the Nash-Williams Lemma, we have
c«(fl. b)m ~ mc.c(A, X - A) - m,4,(A,X -A)"
1 1 '
- j mG{A. X - A) - - ma.(A, X - A) +
+ {*nG.{A, X — A) • irTlv{A,X — A))
> \ (mc{A, X -A)- mc.{A,X - A)) >
> ■= min (mc(A,X - A) - mc-(A. X - A)) -
* Ata
Q.E.D.
Note that if G is a simple 2-cdgc-connectcd graph, then
cG{x. y) > 2 (x. y e X; x ^) •
Consequently there exists a graph //. with the same edges as C. such that
x + y).
In other words, // is strongly connected. This is another proof for Theorem
10.

CONNECTIVITY 185
EXERCISES
• Show that a A-connccied graph is A-edge-connected.
, Show thai a connected graph is 2-connccied if, and only if, for every three vertices
b. *• 'here exists an elementary chain n[a, b] that contains x.
3. Let C be a simple regular graph of degree 3. Show that C is *-connected if. and only if,
C is *-edgc-conn«cied.
4. Tutte has shown that:
' a graph C is 3-connecied if, and only if, G is a "wheel" (an elementary cycle /i and a
vertex Xo joined to each vertex in /0 or can be formed from a wheel by a sequence of
operations of the two following types:
A) the addition of a new edge.
B) the replacement of a vertex x of degree > 4 by two adjacent vertices x' and x" of
degrees > 3 such that a neighbour of x in the original graph is a neighbour of
exactly one of x' and x" in the new graph.
Verify this result with some simple examples.
(Tuttc [1961])
5. A simple 2-connected graph is defined to be mlnlnwllv 2-connecied If the partial
graph rcsuliing from the removal of any edge is not 2-connected.
If C is minimally 2-connected. show that:
A) No edge is the chord of a cycle.
B) If C i* K,, then G contains no triangles.
C) If |.<\ >] is an edge and x is an articulation vertex of C — [x, >■), then x lies on every
cycle that contains [x, >).
D) Fuch chain that contains {x. y] and x contains a vertex of degree 2 in its interior.
(!) If C Kj, there is a cycle of G that contains two non-adjaceni vertices of degree 2.
F) The set of all the vertices of degree 2 is an articulation set.
(M. Plummcr A968))

CHAPTER 10
Hamiltonian Cycles
1. Hamiltonian paths and circuits
In a graph C - {X, U), a hamiltonian path is defined to be a path that meets
every vertex exactly once. Similarly, a hamiltonian circuit is defined to be a
circuit that passes through every vertex exactly once. In a simple graph
G — (X, £), a hamiltonian chain and a hamiltonian cycl'an defined similarly.
Example 1. Voyage around the world (Hamilton). Consider 20 cities
a, b, c /, represented by the vertices of a regular dodecahedron (poly-
(polyhedron with twelve pentagonal faces and 20 vertices). How can we travel to
every city exactly once and return home using
only the edges of the dodecahedron? In other
words, can we mid a hamiltonian cycle in the
graph in Fig. 10.1 ?
Hamilton solved this problem as follows.
When the traveller arrives at the endpoint of
an edge, he has the choice of taking the edge to
his right (denote this choice by R), or the edge
to his left (denote this choice by L), or by
staying where he is (denote this choice by I). In
the obvious way, define a product of these
operations: for example, L2R denotes the
operation of going left twice and then right
once. Finally, two operations arc defined to be equal if after starting from
the same vertex they terminate at the same vertex. Note that the product is
not commutative (for example LR RL), but it is associative (for example
(LL)R - L{LR)). For the graph in Fig. 10.1,
R6 - L8 - I
RlaR -LRL
LR2 L - RLR
RL3 R - La
LR3 L - R* .
186
Fig. 10.1

HAMILTOMAN CYCL S 187
uence,
, - /?» - R2 R3 - (LR3 L) R3 - {LR3J - [I(LR? L) RJ -
- (JLa R3 LRJ - [L*(LR3 L) RLR]* - [JL3 R3 LRLR]* -
- LLLRRRLRI.RLLLRRRLRLR.
This sequence contains twenty operations and contains no partial sequence
that equals I. Therefore, it represents a hamiltonian cycle. Another hamil-
lotiian cycle is obtained by reversing the sequence. It is left to the reader to
verify that no other cycles exist.
Mote that the voyage around the world can begin (and terminate) at any of
the twenty cities. Hamilton solved the problem with the additional constraint
that the first five cities to be visited had to be a, b, c, d, e in that order, lie
found four solutions, starting with edge [a, b]:
RLRLRLLLRRRLRLRLLLRR
RLRLLLRRRLRLRLLLRRRL
RLRLRRRLLLRLRLRRRLLL
RLRRRLLLRLRLRRRLLLRL
'Example 2. Open voyage around the world. Suppose that the traveller in
Example 1 need not return home. In this case, each trip corresponds to a
hamilton chain in the graph in Fig. 10.1, and the number of distinct trips
available to the traveller increases greatly. All the trips starting with an R arc
indicated below. All other possible trips enn be obtained by interchanging the
operations L and R in the sequences given below.
RRRLRLRLLLRRRLRLRL
RLRLLLRRRLRLRLLLRR Rl.RRRLLLRLRRRLLLRL
RLLLRRRLRLRLLLRRRL RLRLLRRRLRLLLRRLRL
RRRLLLRLRLRRRLLLRl
RLRRRLLLRLRLRRRLLL
RLRLRRRLLLRLRLRRRL RRRLRRLRLRLRlLRLLL
RRLRLRLLLRRRLRLRLL RRRLLRlRRRLLLRLRLR
RLRLRLLLRRRLRLRLLL RRRLRRLRLRLLLRRRLR
RLLLRLRLRRRLLLRl.Rl
RRLLLRLRl RRRLLLRLR RLRl.RLLLRRRLRLLRRR
RLLLRLRLLRRLRLRRRL
RRRLLLRLRLRLRRLRRR
RLRRRLLLRLRLRRLRRR RRRLRRLRLRLRLLLRRR
RLRLLLRRLRLLRRRLRL RLRLRRRLLLRRLRlLLR
RLLLRLRRLLLRRRLRLR
RRRLLLRLRLRLRRRLLL
RLLLRLRLLRRRLRLLLR RLLLRLRRRLLRLRLLLR
RLLLRLRRRLLLRLRRRL

188
GRAPHS
Exampie 3. Knight's journey (Eulcr). How can a knight be moved o
chessboard so that he visits each square exactly once? This problem of find ju,
a hamiltoninn chain has interested many mathematicians: e.g. Euler, c
Moivrc, Vandcrmonde, etc. Many methods have been proposed. One method
that seems to work in practice is the following: Move the knight to the square
from which he will control the smallest number of unvisitcd squares.
I'ig. 10.2. Examples of the knight's journey
Another method is to move the knight on only half of the chessboard and
then to have him repeat this pattern on the other half (sec Fig. 10.2). This
method depends on the special structure of the chessboard and does not work
in more general situations.
Example 4. The problem of Mr. No. Mr. No, a mythical Japanese detec-
detective, lives in the upper left square of a chessboard. He wishes to visit Mr. Go
who lives in the lower right square of the chessboard. Mr. No can move to any
adjacent square but cannot move diagonally. Is it possible for him to visit
each square of the chessboard exactly once en route to Mr. Go?
Clearly, the problem of Mr. No is to find a hamiltonian chain between two

HAMU TONIAN CYCLfS 189
ojven vertices. To visit each square exactly once, Mr. No must make 63
moves and 63 colour changes. Clearly, after 63 colour changes Mr. No stops
on a square with a colour different from the colour of his home square. Since
j^r. No and Mr. Go have home squares of the same colour, no such trip is
possible.
Theorem I. The number h(G) ofhami!Ionian paths in a \-graph G ■ {X, U)
has the same parity ax the number of hamihonian paths in the complementary
\.graph G-{X.Xx X - U).
We shall show that if G and C arc two complementary l-gruphs. then
h(G) ■ h(C) modulo 2.
Suppose that the vertices of G are indexed from I to n. Given a subset V of
the arcs of C, let h{V) denote the number or arrangements (/,, /a, .... /„) of
A,2,..., n) such that each are of Vis of the form (/„, /v+1). Note that if h(V)
is not zero, then
m<»- i.
and the arcs of V form a family of pairwisc disjoint paths.
Furthermore, if h(V) 0 and if | I | < />- I, the connected components
of the partial graph (A', V) consist of r disjoint paths, where r > I, and conse-
consequently. /r( V) - r! ■ 0. Hence,
o ~ i n -«- i.
Now note that h(C) equals the number of arrangements (/,, / /„) such
that no arc of G is of the form (/*, lk.\). Thus,
- Y A(V)- I hW) + •■■ + (-!)*"l I h(V) + •••
\v\*i in-' IT*
Therefore
h(G) ■ /;! - /i(C) ■ X h(V) - /.(C).
VCU
I t-|»n-l
Hence, h(C) m h{G) modulo 2.
Theorem 2 (C. A. B. Smith [1946)). In a simpl- regular graph oftlegne 3.
'lie number of hamiltonian cycl s that contain a given edge Is even.
Suppose that there exists a hamiltonian cycle (otherwise, the result is
lrivial|y true). This cycle is of even length (since there arc-/f edges, which
implies that /; is even). Thus the edges of the cycle can alternately be coloured
two colours a imcl (i and the edges not on the cycle can be coloured >'.

190 ORAPKS
Thus, there exists a partition { £0, £,,£,} of the edge set into three perfc
matchings. Such a partition will be called here a 3-colouration of the edgej,.
3-colouration of the edges with three colours a, fi, y determines a vector
Pat ■ (/»!. H* Hm) wi'n /*' ■ 1 if edge e, is a or /?, and /i1 - 0 otherwise.
Let |i e M if vector n defines a family of vertex-disjoint even cycles that ust
all vertices. Each 3-colouration {£., Eg. E,} determines three vectors
. With
H^ + H.» + Pf, ■ 0 (mod. 2).
Furthermore, if a vector n e A/ consists of a family of k(\i) pairwise disjoint
cycles, then it corresponds to 2*"" distinct 3-colourations. Summing the
above identity over nil 3-colouralions { Ea, E«, Ey} gives
Y 2*(")n »0 (mod. 2).
Hence,
V n-0 (mod. 2).
Since this sum is over all the hamilloiiian cycles of C, the number of distinct
hnmiltonian cycles that contain a given edge is even.
Q.E.D.
Corollary I. If a simple regular graph of degree 3 has a hamiltoiiian cycle,
it has at least three hamiltonian cycles.
Let e be an edge of the hamiltonian cycle. At least two distinct hamiltonian
cycles iii and /i3 contain e. Let x denote the first vertex at which these two
cycles diverge after passing through e. If [x, ;■] is an edge of nt and not of n»,
then at least two hamiltonian cycles pass through [x, y), and they arc both
different from ua.
Q.E.D.
Corollary 2 (N. J. A. Sloanc A969]). If a graph G - (X, E) has two liamil-
Ionian cyclis without common edges, it lias at least three hamiltonian cycles.
Let /4t - [.Vi.xa xn, xt] be a hamiltonian cycle of G, and let Ha b6
another hamiltonian cycle or G that does not contain any edge of/i,.
Casi I. Suppose n is even. A regular graph of degree 3 can be formed witn
.ill the edges of>;, and some edges of /<a. Corollary I established thul this graph
has three hamiltonian cycles. Therefore, the result is true for G.
Casf 2. Suppose n is odd and (*,. jc, + aje E for all /. Then, there exists ■
third hamiltonian cycle, nnmcly:
/<3 ~ I'll ■*«» Xl> Xl> X4> •••» •*■—I » A'l] ■

HAMILTONIAN CYCUS
191
Cask 3. Suppose /; is odd and [xn, jca) $ E. Suppose that
Hi m [*/» Xt, Xj, Xkl, Xkl, ..., **„.,, X/] , l,j # M .
Form a new graph C by adding edge (vn, xu] and by removing vertex .v, and
all edges incident to .vt. Graph G has a hamiltonian cycle jtx - [.va, xa
Xn< ■*»)• "lc cd|*s of cycle /Ii and the edges
[.Xj, .V*,], [**,
form a regular graph of degree 3 that has two hamiltoninn cycles that use edge
[xa. -v,,] by Theorem 2. Therefore G possesses two hamiltonian cycles distinct
from /<a-
Q.E.D.
Theorem 3 (Bosak [1967]). A simple regular bipartite graph oj degree 3 has
an eren number of hamiltonian cycles.
Let C - (A', Y, t) be a regular bipartite graph of degree 3 with | X \ -
| Y\ - -', and let h{G) denote the number of hamiltonian cycles in G. For
n < 6, the theorem is true since the complete bipartite graph A.'J>3 contains
exactly 6 hamiltonian cycles.
Assume that the theorem is true for graphs of order < n; we shall show
that it is also true for a graph G - {X, Y, E) of order n > 6. Since
0' contains no triangles, an edge (*,,>',) is adjacent to four edges
[xt.;.,]. 1*1._y3]. [.Vi. Vj). [;',. .v3] such that the vertices *!, xa. xit ylty3,y3
are all distinct.
Let G' and G" denote the two graphs obtained from G by the transforma-
transformations shown in Fig. 10.3.
.'"I
Let h,,h't und h" denote the number of distinct hamiltonian cycles in the
Braphs C, G\ and G" using the vertices jct, xa, xa. yi. ya, y3 as shown in Fig.
l0-4. From the induction hypothesis,
/i(C') - h\ + hi + h'y + h'4 + /.'5 ■ 0 (mod. 2)
h(G") - *; + Wj. + hi + h"A + h"s m 0 (mod. 2) .

192 (JRAFHS
furthermore, it is evident that
Hence,
m h", + h'2 + h\ + h'4
- I *J + I W ■ 0 (mod. 2).
Therefore, the number of hnmiltonian cycles in C is even.
/'.I
''4
/'l
r\
l
v
*',
Q.E.D.
1 1
4
(
(
1 |
1 <
1 1
1 1
1
i(n
10.4
2. Hamiltonlan paths in complete graphs
Theorem 4 (Camion A959]). IfC - (A', T) fa o strongly connected, complete
l-graph, then C has a hamiltonian circuit.

HAMILTON!AN CYCLES 193
Let ft ■ [ai. a-t flji -1. <fy. On * i ■ <*\] he a circuit of maximum length h.
Suppose that circuit ft docs not encounter some vertex />, Then.
a,er{b) btn) W
ber(a,)
Flu. 10.5
Therefore, the vertices not lying on circuit >< can be divided into two classes
B\ and Ba as Follows: If b e Bx, then each arc joining /> and fi is dircctcJ to-
towards /(. If b € Iia, then each arc joining b and >< is directed towards b.
By hypothesis, fl, u ^ 0. Since C is strongly connected, Bx 0,
S3 0, and there exists an arc from Bx to Bx. Denote this arc by (/><,, ^,),
where b3 B2 and bxe Bx.
Thus the circuit [ax, aa an, b a, bx. oj is longer than n, which contradicts
the maximality of ft. OED
Corollary. Lei G ■ (A', f) /«• « compkte \-graph. There exists a vertex x0
Mich that for each vertex y x0 there is a path from x0 to y. Such a tvrwx x0 Is
t illed a "root" ofO. Each root Is the initial eiulpoint of a hamiltonian path ofG.
\. The existence of a root of G was established in Theorem D, Ch. 4), since
any centre of G is a root.
2. Let .v0 be a root of G. Consider the graph G' obtained from graph G by
adding a vertex r, an arc (:,x0), and the arc (x, r) for every x x0. By
Theorem 4, graph C, which is complete and strongly connected, has a
"imiltonian circuit, which corresponds in G to a hamiltonian path starting
from jc0.
Q.E.D.
Algorithm. This corollary provides a very simple algorithm to construct
a hamiltonian path in a complete 1-graph G « (X. I) without loops:

194 GRAPHS
I et ax denote the vertex with the largest outer dcmi-dcgrcc djfli). Verte lfl
is a centre (Theorem 4. Ch. 4). In the subgraph generated by F(ax) - . , I
let oa denote the vertex with the largest outer demi-degrce. In the subgrapi,
generated by r(aa) - {«,, aa}, let a3 denote the vertex with the largest out r
demi-degrec, etc.
Then [a,, a ] is a hamiltonian path.
Theorem 5. IfG — (X, D Is a complete, anti-symmetric, transitirv \-grap
then G has exactly one hamiltonian path.
A hamiltonian path must start from a root. Since G is transitive, vertex*,
is a root if. and only if.
There is always a root in G, and since G* is anti-symmetric, the above equa-
equation shows that the root is unique. Thus, the first vertex of a hamiltonian path
must necessarily be this root xx.
By the same argument, the second vertex of the hamiltonian path is the
unique root of the subgraph generated by X — {xx), etc and the hamil-
Ionian piith is unique.
Q.E.D.
Theorem 6 (R£dci [1W4)). IfG is a complete, antisymmetric \-graph. then
the number h(G) of distinct hamiltonian paths in G is odd.
We shall show that if G is a complete, anti-symmetric graph, then reversing
the direction of an arc docs not change the parity of/i(G). Since graph G can
be obtained from a complete, nnti-symmctric, transitive graph by making
successive reversals of arc directions, this fact will establish the theorem.
Let (a, h) be an arc of G, and let C be the graph obtained from G by re
versing the direction of (a, b). Let G'i be the graph obtained from G by adding
arc (/>, a). Let G2 be the graph obtained from G by removing arc (a, b). We
want to show that
h(G) m h(C) (mod. 2).
G C G, Gt
119
Fig. 10.6

iiamii.tonian rmrs 193
Since graph G2 is unti-syminctric, each hamillonian path of Ga (lukcn in its
reverse direction) defines a hamillonian pulh of Gx, the complementary
I .graph of Ci, and vice versa. Therefore (From Theorem 1), we huvc
/i(C,) - h(C,) m h(G,) (mod. 2) .
Let />>("'>). A,(/>«). /;,@) denote respectively the number of hamiltonian
paths of Gx that contain arc ah, arc ha, neither arc ab nor ba. Then,
/i,@) - h{Gi) m f,(G,) - 11,@) + hx{ah) + hx(ba) .
Thus, >h(ab) m hx(ha) modulo 2, and consequently.
h(G) - /i,@) + lit(ab) m A,@) + hx(ha) - h(G').
This shows that /;((/) ■ h{C) modulo 2, and completes the proof.
Q.E.D.
3. Existence theorems for hamiltonian circuits
Clearly, the greater the dcmi-dcgrccs do(x) and </j(.v) of a 1-graph G are,
the greater arc the chances that G has a hamiltonian circuit. In this section,
we present a theorem of Ghouilu-Houri [I960] that gives conditions for the
existence of a humiltonian circuit in terms of the demi-degrecs of the graph.
First, a lemma is needed:
1 emma. Consider a cinuit with m vertices, each labelled with either a cross
or a circle. Suppose it' contains exactly m0 0 circles and exactly », 0
sequences ofq consecutive crosses. Then
n0 + n, < m - q + 1 .
Suppose that there arc/7 maximal sequences of crosses of cardinality > q.
Each such sequence is framed by two circles. By hypothesis, p > I. Let c de-
denote the total number of crosses. Let a, denote the total number of sequences
°f<7 crosses contained in the i-th maximal sequence. Then the length of this
maximal sequence is? + a, - 1. Hence
c > (q + a, - I) + (q + a2 - 1) + - + (q + a, - 1) >
> q + («i + «i + ••• + «„)- i.
Hence,
«o + «i - «» - c •
Q.E.D.

196
GRAPHS
Theorem 7 (Ghouila-Houri [I960]). Let G be a strongly connected \-grapi,
of order n and without loops. If. for each vertex x.
then G has a hamiltonian circuit.
Let n > 3. Suppose that the result is true for any graph of order < n, and
that there exists a graph G - (X. F) of order | X | - n for which the theorem
fails. We shall show that this leads to a contradiction.
By hypothesis,
(i) |rc*W|+ |rc-(x)|-|X| >0 (xeX).
Let n - [xq, x,, x2 xM_,. x0] be an elementary circuit of C of maximum
length. Since G is strongly connected, m > 2, and since there are no hamil-
hamiltonian circuits, m < n. Let
Xq ~ { Xq, Xj, Xi,..., xm-i}.
Let Xu Xi XT denote the strongly connected components of the sub-
subgraph generated by X - XQ.
I. We shall show that each subgraph GXl. CXu CXf contains a hamiltonian
circuit.
Let xe X,, 1 </</>. Then, for * < m,
(Otherwise, the circuit /< could be lengthened.) Hence,
| TcOO r» <y0 | < | Xo | - | H
ForyeX,,J 1.0,
(because X, is a strongly connected component different from X,). Thus, for
eachy>0.y
n X; | < | X; | - | rc*(x) n X; |.
Mg. 10.7

KAMII TONIAN CYCLES 197
Combining this result with (i) yields
> - I (I /"gOO nXj\ + \ />;(*) nXj\-\Xj\)>0
Since this result is valid for each xe X,, the subgraph 6\, has a humiltonian
circuit by virtue of the inductive hypothesis.
2. We shall show that there exists an X,, i 0, such that
It suffices to show that there exists an X, that is joined in both directions to
XB. Since graph G is strongly connected, there exists at least one X, such that
Gl*o) ft Xj ft 0 .
Consider a vertex x Xo u Xj. If xk e Xo, and if no X, is joined in both
directions to Xo, then
Hence,
<l V V VI I r*~f %• \ >* f V V y \ I
I A ~ Aq •■ A i| ™" I / 0 l^fcJ ■* \** "~ ^ 0 """ ^^'1 •
\(ye X,, then for each x $ Xo u Xh
Hence,
I A — *o — Ay | — | / Q\y) n (X — a0 — Ay)| .
Finally, for eachy XQ\J X,, we have, by virtue of (i),
W | Fa (y) ft (Xo u Xy) | + | Fq (y) n (Xo u Xy) | >
> | Xo u Xy | - | (X - Xo - X;) n ro*(>>) I -
Furthermore, since C Is strongly connected, there exists a path from X, to
*o of the form [r0, r,, h.,:M,;,], with zuz ,r,_i i XQ \J X,. Since
xi ft rs(X0) - 0. this path has length > I.
Consider the subgraph generated by XQ \J X, together with the arc (z0, z,).
Since this graph is strongly connected and has fewer vertices than C, in-
inequality (ii) shows that it contains a humiltonian circuit. This circuit neces-
necessarily contains arc (z0, z(). By replacing arc (z0, r,) with the path r0. rt z,,
wc obtain a circuit of G that is longer than /;, which is a contradiction.

198
ORAPHS
Hence, there is a component X, with the required property. Denote tli
component by A\.
3. Here we shall show that each y Xi satisfies /*£ (}') ^ Xo 0 an
faW^Xo 0.
We may assume | A't | > I, because, if | Xt | - I. this is obviously true.
Let y0 be a vertex of Xi that docs not satisfy, for example,
Let [y^ yt >',_,. y0] be a hamiltonian circuit of Xt with length
2 < q < m. (From Part (I), such a circuit exists.)
Let s be the smallest index such that Faiy,) n Xo 0. From the dclinitio i
of A',, such an s exists, and s 0. )
•vi>
n«. io.8
Since graph C has no circuit of length > m, xke Tot;',) implies that
xk Jf*+,*r^(^,_i). Thus,
| ^clKi-i) n Xo I < m - ij ,
Hence,
| To^.-i) n A'o | + | rg(y,-i) r\X0\-\X0\<m- q + 0-m- -q
and
0, 1,
since there are no double edges between
Hence,
I
-0
) n x, | -
,-> a"d X,.
'.-,) n X, | -

IIAM1LT0N1AN CYCLES 199
which contradicts (i). Hence
r^y) r>Xo + 0 (yeXJ.
By a similar argument,
l'2(y)r\X0 0 (>'€*,).
4, We shall show that, for each y,eXt,
| rSiy.) nX0\ + \ rc+(>',_,) nX,|<m-? + l.
In Tact, Xo is a circuit of m vertices that can be labelled with circles and
crosses in the following way: If x, e r<- (yt. x), label x, with a circle.Otherwise,
label x, with a cross.
We have seen in Part 3 that xH F~ {y,) implies the existence of a sequence
•**+!• xk+lt — > •** + «
of g crosses. From 3, there arc n0 0 circles and h, 0 sequences of q
crosses. From the lemma,
| y.) n Xo | + | r^y.-O n Xo \ < n, + n0 < m -
5. W» a7ki// show thai there exists a vertex y,sX{ with
From inequality (i). each vertex .y A'i satisfies
|rj(r)nxo| + |re-(y)nxo|>
> Uo I - (I rG+W n *, I + I TJOO r> *i I - I *i I)
I {j jj
J*0.l
> m - [(</ - 1) + (<j - I) - <i] - 0 - m - q + 2 .
By counting in two different ways the number of arcs joining Xo and A'i, we
obuin
, X0\ + \ rc*(>f.,) n Xo |) -
- I (I ro*(^.)n Xo I + I rc"(^)n Xo I) > q(m - q + 2) •
1-0
Therefore, there exists at least one y, that satisfies the above inequality. This
contradicts Part 4.
Q.E.D.
Corollary 1. // C ■ (X, D is a \-graph without loops such that
dc(x) + t/J(x) > n - 1 (xeX),
thin G has a hamiltonian path.

200 GRAPHS
Add to G a vertex x0 and join it to each other vertex by two oppositely
directed arcs. The new graph C is strongly connected, and
drt-to + «£•(*) > n - 1 + 2 - »' (x e X').
Graph C has a hamiltonian circuit from Theorem 7. Thus graph C has
hamiltonian path.
QE.D.
Corollary 2. If C is a complete \-graph, then C hits a hamiltonian path.
(This result follows also from the corollary to Theorem 4.)
Corollary 3. Let C - {X, /") be a strongly connected \-graph of order n
without loops. If the graph remains strongly connected after the removal of any
vertex, itnd if
doM + da(x) > n + 1 (xeX),
then for each pair a, b of distinct vertices, there exists inC a hamiltonian path
with endpoints a and b.
Consider the graph C obtained from C by contracting {a,b} into a single
vertex c and by letting
/~c+.(c) - r*{b) -{a}.
Graph C is strongly connected because for x a,b graph C has a path
from x to (i that avoids b, and a path from b to x that avoids a. Thus, C hasa
path from .v to r, and a path from c to x.
These properties hold also for the graph C" obtained from C by contracting
{a, b) into a vertex c and by letting
r£-(c) - r£(a) -{b},
ro-(c) - r~(b) -{a}.
Furthermore,
I rfrc) | + | rr;.(c) | + | r^c) \ +1 rc- @1 >
> | rc+(b)| -i + | rr»| - l + | r*(a)\ - 1 +1 ro-(W| '
>2|X|-2-|X'| + |X'l-
Thus at least one of the following two inequalities is satisfied:
rc+.(c) | +1 roic) I > I x11,

HAM1LT0N1AN CYCLES 201
Suppose, for example, that the first inequality is satisfied. Then graph C
satisfies
drt-M + da-M > | X | - 1 - | r | {xeX').
Thus, from Theorem 7, C has a hamiltonian circuit that corresponds in G to a
hnmiltonian path between a and b.
Q.E.D.
Corollary 4 (Nash-Williams [1969]). // C - (AT. T) is a \-graph without
loops, of order n, such that
d^)>~, d-0{x)>~ {xeX),
then C has a hamiltonian circuit.
From the Ghouila-Houri theorem, it suffices to show that C is strongly con-
connected. More precisely, we shall show that a I-graph G without loops with
d£M > ™-i , daM > ~- {x e X)
is strongly connected.
Suppose that C were not strongly connected. Then there exist in C several
strongly connected components Xx, Xa Xp, and their contraction yields a
graph without circuits (Theorem 12, Ch. 3). There exists at least one connected
component Xt wiih/Ho(A'1, X - Xt) - 0, and one connected component X3
with mr;(Af- X3,X2) -0.
Suppose, for example, that | Xt \ > | X31. If .voe X2, then
This contradicts the hypothesis.
Q.E.D.
Corollary 5 (Bcrmond [1970)). IfG ■ {X, D ha \-graph of order n without
l°°ps such that
> --ti , ,/J(x) > !±± (x e X)
is an integer, 0 < A: < n - 1, r/jf« foc7i elementary path of length k is
contained in a hamiltonian circuit.
Let u0 - [a0, <j, aK) be a path of length k in C. Let
A - { o0. «i fl*} •

202 GRAPHS
Construct a I-graph Co from C by removing A, adding an auxiliary vertex
adding an arc (x, a) Tor each * e X — A with x e Fated, and adding an are
(a, y) for each j» e X - A with y e /"£(a*).
It suffices to show that <70 has a hamiltonian circuit. Graph <70 has n0
n - k vertices. For a vertex * a.
'"c(*. A - { a0 })
Furthermore,
rfo» - rfo(fl») - «5(fl*. /» - { fl*})
n\c(ak, A - {ak})> do(ak) - k
Thus, for each vertex x of graph Co,
n-^r -k - t ■
Hence, from Corollary 4. Go has a hamiltonian circuit, that corresponds in C
to a hamiltonian circuit containing /;0. n E D
Corollary 6. If C - (AT. /') « fl \-graph without loops cf order n such that
i (
then for each pair a, b of distinct vertices there exists a hainlllonian path from
a to b.
If a r(b). let C m C; otherwise let C - C + (h, a). Graph C satisfies
the above conditions. Therefore, from Corollary 5. C has a hamiltonian
circuit contuining (b, a) that corresponds in G to a hamiltoninn path from a to
/>• Q.E.D.
Recently, the following generalisation of Theorem 7 has been proved:
Let G be a strongly connected I -graph of order n with no loops. If every
pair xy of non-adjacent vertices satisfies
rff)(.v) + 4J(x) + </,» + do(y) > 2n - I,
then G has a Immillonian circuit (Mcynicl [1973]).
We may also mention the following conjecture due to Nash-Williarns
[1969].

HAMILTONIAN CYCLES 203
Conjecture .ifGisa l-graph without loops of order n > 5 such that for all
li <. s. the number of vertices x with d^lx) Kk is < k, and the number of
ftrtices y with </J(v) < k is < k. then G has a hamiltonian circuit.
When sufficient conditions for the existence of a hamiltonian circuit are
difficult to find, necessary conditions can be found by using the concept of
a dissection.
A dissection of graph G is defined to be a set of elementary paths of G such
that each vertex of the graph is contained in exactly one path. Call a path
closed if it also defines a circuit, otherwise call it open.
The value of a dissection is defined to be the number of open paths in the
dissection. The value is always less than or equal to the number n of vertices
since a single vertex is a path of length 0. A hamiltonian path is a dissection of
value I. A hamiltonian circuit is a dissection of value 0.
Dissection theorem. For a \-graph G - (X, I), the minimum value of a
dissection equals
So - max (| S | - | f(S) |).
let
Consider a dissection a - (a1, a3 a", fil, fi2 ff) of value q, where «(
is u closed path, and /?' is an open path. Let A' denote the set of all vertices
a1, and let B' denote the set of all vertices in fi'.
Makctwocopics A"and Jfof A", and form the bipartite graph // - (A\ Jf. E)
where (*,, x,) 6 E if, and only if, x, T(x,).
Lacli circuit a1 defines uniquely in // a matching of | A' \ edges, and each
open path defines in // a mulching of | B' \ — I edges. Therefore the dissec-
dissection defines a matching £0 of cardinality
Thus
iIlI
< J i I
This correspondence between the matchings and the dissections is a bijec-
l'°n. Therefore, from the Kttnig theorem (Chapter 7). the minimum value of a
^section equals
n - max \E0\ - n - (n - So) ■ So .
•his gives the required formula.
Q.E.D.
Corollary 1. If G has a hamiltonian circuit, then 60 - 0.
The proof is immediate.

204 GRAPHS
Corollary l.IfC has a hamUtonian path, then 0 < <50 < 1.
The proof is immediate.
Corollary 3. If a \-graph has no circuits, then a necessary and sufficit i
condition for the existence of a hamiltonian path is that <50 - I.
The proof is immediate.
4. Existence theorems for hamiltonian cycles
Without loss of generality, we shall assume throughout this section that ;
is a simple graph.
The main existence theorem for a hamiltonian cycle (P6sa [1962]) was
generalization of a previous result of Dirac A952]. A new proof for P6st'i
theorem, due to Nash-Williams, permitted J. A. Bondy A969) to give
stronger result. Finally, Chvtltal A972) has proved an even stronger thcorc,
that is in some sense the best possible. The following theorem is an extensio
of Chvatal's theorem that has been modified to generalize a result of Kronk
[I969J.
Theorem 8. Let G - (X, E) be a simph graph of order n with degr
d\ < </a < ■" < dn. Let q be an integer. 0 < q < n — 3. If, for evei •
with q < k < \ (« + </), the following condition holds'.
(A) rfv_, < k •> dn.k > n - k +q
then for each subset F of edges with \ F \ - q such that the connected c
ponents of(X, F) are elementary chains, there exists a hamUtonian cycle
that contains F.
Furthermore, this residt is the best possible in the following sense: each
quence of degrees that does not satisfy condition {A) is majorizedby a seque>
of degrees of a graph that does not have the desired property.
I. We shall first show that condition (A) implies: </, > q.
Suppose that di < q, and let k - q + I. Then
Furthermore, since </, < k, it follows from condition (A) that
</„-,., > n - q - I +q - n - I .
This shows that r xist at least q + 2 vertices joined to all the olh
vertices of C; th cforc dt > q + I, which is a contradiction.

IIAMILTON1AN CYCLES 205
2 We shall show that if a simple graph G satisfies condition (A), the
raph G' obtained from G by adding a new edge also satisfies condition (A).
S't-ixJdoMtk}
Clearly. I ■$* I < I Sk |. Note that condition (A) is equivalent to
(A-) \Sk\>k-q •> | Sn. „♦,.,!< n- k.
Since G satisfies (A1),
|S|>*-? » |s*|>*-? » is,.,.,.,! <»-* «>
* I •£!-*♦,-I | < « - A.
This shows that C also satisfies condition (A).
3. To prove the main part or the theorem, suppose that there exists a
graph satisfying condition (A) that docs not have the desired property. By
adding new edges as many times as needed, we obtain a graph G - (X, E)
satisfying condition (A), that docs not have the desired property, and such
thai the addition of any new edge gives a graph with the desired property.
Let F <= E be a set with | F \ - q that forms pairwise disjoint elementary
chains, and that is not contained in any hamiltonian cycle of G.
Since G is not complete, there exist two non-adjacent vertices ^ and yn;
choose yt and yn with ilj,)'i) + rfo(>'n) as large as possible and with
There is a hamiltonian cycle n' containing the edges of F in graph C -
c + (JWnli and« necessarily. [yi,yn] -;<': therefore, there is a hamiltonian
chain n in G that contains F, of the form
(i - [yi.y y*\-
Let
/-{//I </<«- U>w.M]e£,!>
*ct / is not empty, since
(B) I /1 > rfoO-i) - q > rfi - d > 0 .
Moreover,
<C) is I •> [y,,yME
c, otherwise, the cycle
°uld be a humillonian cycle containing F.

206 ORAPHS
4. Let * ■ d<Ayt). We shall show that
(D) *<*<^±i.
Since dt > q, the first inequality of (D) holds. From (C), it follows that
Hence,
do(yn)<i(n- I)-
Thus
(E) dJiyY) + do(yH) < n + q - 1
Finally,
. ... n +q
* - doO-i) < —y~ •
5. From (Q, we know that, for each ie I,
[yi,y*\ e
and
Thus,
(/e/).
Since | /1 > A - 9, there arc k - q vertices of degree <k. Hence
which implies that
dn.k > n- k + q.
This shows that there are at least k + 1 vertices of degree > n
Hence, for some x >',,
1'[]*£
I /() * + q
Thus,
« + q
This contradicts (E) and completes the proof of the theorem.

HAMILTONIAN CYCLFS 207
0 To show that the conditions of the theorem arc the best possible, con-
a non-decreasing sequence </, < dt < ■•• < dH and an integer k such
that
n + q
Sequence {(I,) is majorized by sequence ((/O, where
(k \f 0 < i < k - q
n - k + q - 1 if k-q<t*n-k
n - 1 if n - He < f < n .
We shall show that there exists a graph C with the sequence (<i\) for its
degrees. Let the vertex set of 6" be the union or three disjoint sets A, B, C,
where Aha set Sk_, of A- - q isolated vertices, B is a clique Kk of A- vertices,
and C is a clique ^,-ajt,, of n — 2k + q vertices. Join each vertex of B to
each vertex of A u C. Graph C has order n, and da-(a) ■ k if o e ,4, da-(b) -
n - 1 if b e B, and do(c) mf,-k+q-\ifc C.
Consider a set /■"of edges of C that forms an elementary chain of length q
in B. and suppose that there exists a hamiltoniiin cycle /(that contains F. In
sequence /«, at most it - q elements of B arc followed by an clement of AvC.
Since | A \ - k - q, there are exactly k - q elements of B followed by an
dement of A. Thus there is no element of B followed by an element of C,
which is impossible, since \C\-n-2k+q>0.
Q.E.D.
Corollary 1. Lei G be a simple graph with degrees dy < da < •■• < dK. Lei
9 be an integer, where 0 < q < n - 3. If the following condition holds
n, for each set I-of edges with \ F \ - q stuh that the components of{X, F)
ore vertex disjoint elementary chains, there exists a hamiltonlan cycle that con-
'i F.
Suppose that G is a simple graph satisfying this condition. We shall show
that the graph G also satisfies the conditions of Theorem 8.

208
GRAPHS
If the conditions of Theorem 8 were not satisfied, there would cxi. n
integer k such that
q < k <
n + q
dk., < A;
</„.» in-k + q-l.
Let i - A - ?and / - « - A; then / < y since, otherwise, A
we have
1 < / < J < «
■ 4 < / + 9
and, consequently,
dt + d, > n + q.
On the other hand, we have also
which yields a contradiction.
n + q
Thus,
Q.E.D.
Corollary I was proved directly by Bondy [1970]. It has been generalized
by Las Vcrgnas [1970], who proved the following stronger result:
Let C - (X, E) be a simple graph of order n > 3. Let the vertices x, ofC
indexed arbitrarily, and let q be an integer, where 0 < q < n - I. If
I</<;<«
l+J>n-q
d<Axt) < / + q
d<Axt) Zj + q- \
[xt,x,)iE
+ da(xf) > n+q
then, for each f «= E with \ F \ - q such that the connected components
(X, F) are elementary cycles, there exists a hamUtonian cycle that contains
The following result is an easy consequence of Corollary 1.

HAMIITONIAN CYCLES 209
Corollary 2 (Kronk [I969J). Let G be a simple graph of order n, and let q be
Integer, where 0 < q < n - 2. Suppose that
i\) for each k such that q < k < \(n + q - I), the set Sk of vertices of
legree < k has cardinality < k — q,
B) if" + <i ls otltl< and ifk -\(n + q - \), then | Sk | < k - q.
Then for each set I' of edges with \ F \ - q such that (X, F) consists of vertex
disjoint 'lemintary chains, there exists a hamiltonian cycle that contains F.
If the degrees of G arc indexed in increasing order, conditions (I) and B)
arc equivalent to
(I') dk.q>k if
B') </*-,♦! > k if k--(n + q-\).
We have dk.n > k for all k < " + * " ' from (lr). For k - " + ^ " '.
we have from B'):
1
d,-k - </*-.+1 >2^n + 9+ 0 -«-*+?.
In cither case, the conditions of Theorem 8 are satisfied, and there exists a
hamiltonian cycle that contains F.
Q.E.D.
Corollary 3 (Chvdtol [1971]). Let G be a simple graph of order n > 3 with
degrees di < d3 < - < t/R. //
t/k < fc < - » dH.H > n - k,
'hen there exists a hamiltonian cycle in G. Furthermore, if the sequence (</,) does
"°i satisfy the condition, there exists a graph G', with a sequence of degrees
"tojorizing (d,) that has no hamiltonian cycles.
This Corollary is established by setting q - 0 in Theorem 8.
Note that Corollary 3 does not characterize all the sequences that imply the
existence of a hamiltonian cycle. For example, Nash-Williams has shown that
«ach regular graph of degree h with 2 h + I vertices has a hamiltonian cycle,
and the corresponding sequence does not satisfy the above condition.

210 GRAPHS
Corollary 4 (Bondy A969]). let G be a simple graph of order n > 3 . .
degrees dx < da < ••• < </„• If
[ d, + d,> n,
then G has a hamiltonian cycle.
This corollary is established by setting? ■ 0 in Corollary I.
Corollary 5 (Chvatal [1972]). Let G - {X, Y, E) he a simple bipartite graph
such that | X \ - | }' | - » > 2. J
//»£■« Cr An* o hamiltonian cycle.
Let G' be the graph obtained from G by joining each pair or vertices in Y.
Since a hamiltonian cycle of G' cannot contain an edge of 0" — G, it is
sufficient to show that graph G' has a hamiltonian cycle.
The degree sequence of G' is dk ■ do(xw) for it < n, and d'k -
^o(rfc-n) + (« - 1) for A: > 'i. Clearly, this degree sequence satisfies the con-
conditions of Corollary 3 and. consequently. G has a hamiltonian cycle.
Q.E.D.
Corollary 6 (Chvatnl [1972)). If </, < d2 < •» < </„ ore //«• ^r«-5 of a
simple graph G of order n > 2, o/irf if
*< ft - 1 <!■- 1 » rf,.,-* >n-*
//(f« C /iG.v a hamiltonian chain.
Let C be the graph obtained from G by adding n vertex xH. i and by joining
this vertex to all the vertices of G.
Let (</,') be the degree sequence of G'\ for k < (w + ') ~ *, wc have
</;-</*+!<* » f/;M.n ■ t/,*!.* + i > /> + i - ft.
From Corollary 3, graph C has n hamiltoninn cycle, and therefore,
G hns a hamiltoninn chain.
Q.E.D

IIAMILTONIAN CYCI ES 211
li is easy to verify that this result is the best possible in the same sense as
Theorem 8. Besides, Las Vergnas [1971] has extended this corollary to prove
• sufficient condition for the existence of a spanning tree of G with ut most h
pendant vertices.
The following corollaries arc classical results:
Corollary 7 (P6sa [1962]). Let G be a simple yraph of order n > 3, such that
A) for each k such that I < k < —-—, the number of vertices of degree
< k is < k,
B) (///i is odd) the number of vertices of degree < —r— is < —-—.
Then G has a hamiltonian cycle.
This corollary is established by setting <y ■ 0 in Corollary 2.
Corollary H. 1 et G be a simple graph of order n > 3 with degrees dv <
rfj < — < (ln- Ifk < r implies dk > k. then G has a hamiltonian cycle.
Since k < implies dk > k. the number of vertices of degree < k is
< A-. From Corollary 7, G contains a hamiltonian cycle.
Q.E.IX
('orollnry 9 (ErdOs, Gallai [1959]). Let Gbea simple graph of order n > 3.
Ut .v, be the vertex of minimum degree. lfdo{xx) > 2.andifd\x) > -s for all
x ¥> xx, then G has a hamiltonian cycle.
If the vertices are indexed as in Corollary I. we have do(xx) > I. For
I < k < -, we hnve dc(xK) > - > k. From Corollary 8. there exists a hamil-
'onian cycle.
Q.E.D.
Corollary 10 (Ore A961]). LetG - (X, E) he a simple graph of order n > 3,
that
+ do(y) <n => [x, y] e E.
G has a hamiltonian cycle.
:t k < ~. From Corollary
'*/xe X, da(x) < k ) has cardinality < k.
Let k < -. From Corollary 2, it suffices to show that the set Sk -

212 GRAPHS
From the hypothesis. SK is n clique because * < =• Also | Sk | < k i
since the degrees of its vertices arc < k.
| Sh | fc + I because, otherwise. i>\ would be a connected component of
C. nnd two vertices xeSk and ye X - S* would not be adjacent, implying
dc(x) + </oO-) < * + (/i - * - 2) - /i - 2 < /i.
This contradicts the hypothesis or the corollary.
| S,, | A; because, otherwise, euch vertex of Sk is adjacent to at most one
vertex of X - Sk, and the number of edges leaving Sk is
moEk. X - Sk) < | S* | - * < ^ .
Since | A" - Sk | > «< this implies that there exists a vertex y X — Sk non-
adjacent to SK. Let x e Sfc. Vertices .v and y are non-udjaccnt, und
d<Xx) + do(y) <* + (n-*-l)-/i-l</i.
This contradicts the statement of the corollary.
Thus \ Sk\ < k, and G has a hamiltoniun cycle.
Q.E.D.
Theorem 9 (Ore [ 1961 ]). Let C be a simple graph with n vertices and m edges
such that
'«>»(«- 1)(ii-2) + 2.
Then C has a hamiltonian cycle. Furthermore, this inequality is the best possible.
1. From Corollary 10, it suffices to show that there do not exist two non-
adjacent vertices a and b with
dc(a) + dc(b) < n - 1
If two such vertices exist, graph C can be obtained from the complete
graph Kn by removing at least
[(/( - 2) - dc(a)] + [(« - 2) - rfc(A)] + I -
- In - 3 - [</0(<7) + </ctf>)] > 2/1 - 3 - (« - 1) - n 2
edges. Therefore, the number m of edges in C satisfies
m < i ii(fi - 1) - (
which contradicts the hypothesis.
m < i ii(fi - 1) - (ii - 2) - i (ii - IX" - 2) + I.

MAMILTONIAN CYCL S 213
2. Construct u graph Go with n vertices by taking a clique Kn.x with n - I
vertices and adding u vertex x0 that is joined to AVi by a single edge. The
nUniber of edges in Co equals
w(C0)-i(M- l)(/i -2)+ I .
Graph Co hus no hamiltoniun cycles. This shows that the inequality of the
theorem is the best possible.
Q.E.D.
The following theorem is a slightly stronger formulation of a result due to
Erdos and Chvdtal.
Theorem 10(Erdtis, Chvdtal A972]). ictG - (X, E) be a simple graph, and
lei h and q be two integers such that /? > 2 and 0 < q < 2. If graph G is h-
connected, and ifG contains no stable sets o/h — q + 1 vertices, then, for each
f c E with | F | < q. G has a hamiltonian cycle that contains F.
Recall that u set of vertices is called stable if no two of its vertices arc
adjacent. Let Fbc a set of edges with \F\ < q. From Corollary 4 toTheorcm
C, Ch. 9), there exists a cycle ft that contains F. Assume that ft is the longest
cycle with this property. Then, | >/1 > h + 1 because Corollary 4 to
Theorem C, Ch. 5) shows the existence of u cycle that contains any A — 2
vertices and any two edges not incident to these h — 2 vertices.
If ft is not a hamiltonian cycle, there exists a vertex a $ ft. Since | fi \ > h.
Corollary 3 to Theorem C, Ch. 9), shows that a can be joined to the vertices
otft by h elementary chains /<,[<7, *,], fa[a. x3],..., fi>,[a, xh], that are pairwise
vertex-disjoint (except at a). Let x, denote the unique vertex of/(, in ft, and let
>i denote the vertex of/< that immediately follows .v, in ft.
Let / - {// [a-,, y,] $ F). No yt, let, is adjacent to a, because, otherwise,
there exists a cycle longer than /<. Since | /1 > h - q, and since the set
'}',IIef}\j{a} cannot be stable (its cardinality is > A - q + 1), there is un
cd6e [JW/I ''n G w'»» Ue / and / < /
Consider the cycle
t< xt) - n,[a, x,} + n,[a, xt] - p[.v,, y,] + [y,,y,}
'hat contains F. The length of this cycle is greater than the length of /j. This
^ntradicts the maximality of >*.
Q.E.D.
Remark. The bound h - q + I on the cardinality of a stable set is the best
j"J*siblc: For q - 0. the complete bipartite graph A^.am is A-connected and
*s no hamiltonian cycle. For q - I, the complete bipartite graph Kh,h

214 GRAPHS
augmented by one edge (between two vertices or the first class) hus no harni|L
Ionian cycle that contains this additional edge. Another example is th,
Petcrscn graph (Fig. 10.11).
For q - 2, h > 3, a counterexample is provided by the graph K^iK Ug>
mentcd by two edges joining vertices of the first class und one edge joining
two vertices or the second class. No hamiltonian cycle cun contain the first
two additional edges.
One can conjecture:
Conjecture. Let G - (X, E) he a simple graph with connectivity k(G) and
with stability number a(C) such that <x(G) < k(G") - <y. Let F c £ he a set
edges with \ F\ — q such that the connected components of (X, F) are /.
memory chains. Then G has a hamiltonian cycle that contains F.
by Theorem 10, this conjecture is true Tor q - 0. 1.2.
In the following results, we shall only consider simple bipartite graphs..
Theorem 11 (Las Vcrgnas [1970)). let G - (X. Y. E) be a simple hipartit
graph, where X - { *,, .va,.... xn}, Y - {)\, y2 yn), n > 2: let q be n
Integer. 0 < <? < n - 1. //
(A)
do(yk) Z k +q
+ do(yk) > n + q + i
then each set F of q edges that forms vertex-disjoint chains is contained in a
hamiltonian cycle ofG.
Suppose that there exists a bipartite graph that satisfies condition (A) but
docs not have the required property. By adding as many new edges to this
graph us needed, we obtain a bipartite graph G « (X, Y, E) satisfying
condition (A), but not the required property, und such that the addition f
uny new edge gives a graph with the required property.
Lot F c £ be a set of q edges that is not contained in uny hamiltonian
cycle of G. Lclj(x) denote the index of vertex x X, and let k(y) denote the
index of vertex ye Y.
I. Let a, eX und 6, e Y be two nonadjacent vertices that
J{at) + k(b,). The graph obtained from G by adding edge [altbx] has •
hamiltonian cycle that contains Fund, therefore, gruph G has a hamiltonian
chain (i that contains F. Let
H - [altbi, aa, b3 a,.,, b fl,.*i].

HAMILTONIAN CYCLES 215
2. Let
Then. | / I > <W«i) - ?. If/£ /. then [a,_,, *,] £ £ because, otherwise,
[fli, 6,,fl(, 6,n,..., a,, />,, fl,.,, ft,.,,..., fra, a,]
would be a hamiltonian cycle containing F. Thus
r^Mc X- {a,.,//£/}
and so
or
(B)
3. Lety(«i) ■ J, k{h1) - ?, so that ax - *„ *, - y,.
If r e /, it follows from Part 2 that [a,.,, A,] # £, and. therefore,
Hence
M-i) < s.
Thus there arc at least | / | vertices x with^A-) < j, and, therefore,
x > | /1 > rfG.(Wl) - 9 .
Hence
('cW ^ * + a .
By the same argument,
doM < / + q.
Since C satisfies condition (A), and since [*,, >',] ^ £, it follows that
d(Ax,) + da(yt) > n + q + 1 .
This contradicts (B).
Q.E.D.
Corollary 1 (Bondy [1969]). Let G - (X, Y, E) be a simple bipartite graph
with | X | - | Y | - n > 2. // the vertices of X and of Y are indexed with
Creasing degrees, and if
+ da(yk) > n
'hen G has a hamiltonian cycle.
The corollary is established by setting / - 0 in Theorem 11.

216 GRAPHS
Corollary 2. Let G - {X, Y, E) be a simple bipartite graph with \ X \
| Y | - n > 2, am/ /?r 9 6c an Integer, where 0 < 9 < /», jwc/i r/wr
(I) for each integer j < 2-±-?, /to jm S, - {x/x€ X, do(x) *j) fa
cardinality < j - q,
B) for each integer k < n—^-, the setTK-{yfye Y, ddy) < k) has
cardinality < k — q.
Then for each set Fofq edges that forms vertex disjoint elementary chains,
there Is a hamiltonian cycle that contains F.
This follows easily from Theorem II.
Corollary 3 (Moon, Moscr [1963]). Let G - (X, Y, E) be a simple bipartite
graph with \ X \ - | Y \ - n > 2. If\ S* | < k and \Tk\ < k for each Integer
k < — ■ Then G has a hamiltonian cycle.
The corollary is established by letting q - 0 in Corollary 2.
Corollary 4. Let G - (A\ Y,E) be a simple bipartite graph with \ X\ -
I Y I - n > 2. // f flr/i rcr/f Jf z G A' u K satisfies djj) > - , except for two
vertices, *,e X with d^Xi) > 2, andyxe Y with da{yv) > 2, then G has
hamiltonian cycle.
If A — I, then
|Sk|-0<*; |rk|-0<*.
If* > I and* <~< then
I Sk I < I < k ; I Tk I < I < k .
Corollary 3 can now be used to show the existence of a hamiltonian cycle
in G.
Q.E.D.
Corollary 5. If, G - (X, Y, E) Is a simple bipartite graph with \X\m
I Y I - n > 2 such that
rfG(.v) + duiy) < « » [*, .>']€£•,
f/i«i C /kt.t fl hamiltonian cycle.
The corollary is established by letting9 - 0 in Theorem II.
Corollary 6. let G - (A', K. £') Ac o $//»/>/? bipartite graph with
I A'I - I Y\ - «. I £| - »i.

HAMILTONIAN CYCLES 217
m > n1 - n + 2,
then C has a hamiltonlan cycle.
It suffices to show that there do not exist in G two nonadjacent vertices x
8nd y with da(x) + d^y) < n. If two such vertices exist, then graph G can be
obtained from the complete bipartite graph A'r, „ by removing a set of edges of
cardinality at least
(n - deW) + (" - <*e(y)) - 1 - 2 « - K(*) + 4,0')) - I > n - I .
On the other hand, the number of edges eliminated from Kn,n is
n2 - /w < «2 - (n2 - n + 2) - n - 2,
which is a contradiction.
Q.E.D.
5. Hamilton-connected graphs
Consider a simple graph G - (X, E). Graph G is defined to be Hamilton-
connected if for each pair x, y of distinct vertices, there is a hamiltonian chain
with endpoints x and y. In this section we shall study sufficient conditions for
a graph to be Hamilton-connected.
The above definition can be generalized. Let F be a set of q edges that form
vertex-disjoint elementary chains. Let x, y be a pair of distinct vertices which
are the endpoints of two distinct chains of (X F). Graph G is defined to be
q-lhwiihon-connected if for each such F and for each such pair x, y, there is a
hamiltonian chain with endpoints x and y that contains F.
For q - 0, a ^-Hamilton-connected graph is a Hamilton-connected graph.
a. For two integers q and n with 0 < q < n - 2. let Jf(n.q) be a
(lass of simple graphs of order n satisfying the two following conditions:
A) IfG e Jf(n, q), each set Fofq edges forming a system of vertex-disjoint
elementary chains Is contained In a hamiltonian cycle.
B) IfG e. (n. q), the graph G' obtained from G by adding any new edge also
belongs to (/», q).
Then each graph of. (n. q) is (q - \)-llamilton-connected.
L« G - (X. E) be a graph of. (n, </). and let F be a set of q - 1 edges of
G such that the connected components of (X, F) arc elementary chains. Let
*•>" be two vertices that arc the endpoints of two distinct chains of {X, F).
"" [x, y] e E, let G' - C: otherwise, let G-G + [x, y].

218 GRAPHS
From condition B), graph G' belongs to . (n.q); since Fu{ [.v,y)}
stitutcs a system of vertex-disjoint elementary chains of total length q,
G' has a humiltonian cycle that contains F\j {[x.y]}. Therefore, F is <.
taincd in a humilloniun chain of G with endpoints x and y,
QE.D.
Theorem 12. Let G he a simple graph of order « > 3 with degre j
d\ < d% < ••• < dn that satisfies the following condition:
«/*-, < k
- , n
Then graph G is Hamilton-connected.
The graphs that satisfy this condition form a class («. 1) by Theorem
Q.E.D.
Ihcorcm 13. Let G be a simple graph of order n > 3 wch that
A) for each Integer k, where I < k < ^, the set Sk of vertices of degree ,
has cardinality \Sk\ < k — 1,
B) ifn is even, \ Snla \ < '± - I .
Then, graph G Is Hamilton-connected.
The graphs that satisfy these conditions form a class. (n, I), by Corollary 2
to Theorem 8. Q.E.D.
Theorem 14. Let G - (X, Y,E)bea bipartite graph, with \ X \ - | Y \ •
n > 2, In which the vertices xte X andy, e Y are Indexed such that:
Suppose that, if} and k are the smallest two indices such that
(if they exist), we have d^x,) + </«(>■») > n + 2.
Then, for every pair xy, xeX, ye Y, there exists a hamiiwnian chain
endpoints x and y.
From Theorem II, the graphs that satisfy these conditions form a class
B w.l).
Q.E.D.

HAMILTONIAN CYCLFS 219
fhcorcm 15. Let G - (X, Y, E) be a bipartite graph with \ X | - | Y | -
a 2 satisfying the two following conditions:
A) /or «fc/iy < j " • 'Ae set °f vert'ces xt of degree < y A<m cardinality
<J- I.
B) /«"" «<"■* * < - j— . the set of vertices y, of degree < * Aoj cardinality
<k - 1.
'* Hamilton-connected.
From Corollary 2 to Theorem II. the graphs that satisfy these conditions
form a class. B«. I).
Q.E.D.
Theorem 16 (ErdCs, Gall li [1959]). Let G be a simple graph of order n > 3
Mth do{x) + da(y) > n far each pair x. y of distinct, non-adjacent vertices.
Then, graph G is Hamilton-connected.
From the Lemma, it suffices to show that
A) x f* y. </,.(*) + da(y) < n - [.v, y) e E
implies
B) each edge of G is contained in some hamiltonian cycle.
The proof is similar to the proof for the first part of Theorem 8. Let G be a
simple graph that satisfies A), but not B). We may assume us before that the
addition of any new edge causes the graph to satisfy property B).
Let e0 be an edge of G that is not contained in any hamiltonian cycle. There
wist two non-adjacent vertices a and h, since G Kn. Since G + [a, b] has a
hamiltonian cycle containing e0, there exists in C a hamiltonian chain con-
containing e0 that is of the form
l<[a, h] m la.yt.ys .»'■-!.*]■
Lcl >•! - a, and let
/-{///> 2. [a.y,] E, [y,.,.y,) * e0}.
Then.
F°r /e /. we have yt.xi Ta(h) because, otherwise.

220 GRAPHS
would be a hamiltonian cycle containing e0. Thus,
n- I -do{b)>\I\>dc(a)- I
and
do(h) < n .
But, from (I), this implies that [a,b]e E which is a contradiction.
Q-E.D.
Theorem 17 (Ore [1963]). If G is a simple graph of order n with m edges such
that
then G Is Hamilton-connected.
Let G - (X, E) be a graph thut satisfies the inequality. Consider two non-
adjacent vertices a and b. Let Eo denote the set of edges adjacent to neither a
nor b. Then,
and
Thus,
<U«) + </c(fc)-|£| -|Eol >
> -7"[(" - I) - (n - 3)] + 3 - « + 1 ■
Therefore, from Theorem 16, G is Hamilton-connected.
Q.E.D.
The inequality presented by this theorem is the best possible: Consider the
complete graph #,, with n vertices and remove all the edges incident to vertex a
except for two edges.
If n > 4, this graph is not Hamilton-connected because the two verti
adjacent to a cannot be the endpoints ofa hamiltonian chain. The number of
edges of this graph equals
wi - i n(n - |) _ (« _ 3) - I (« _ |) („ - 2) + 2
Theorem 18 (Moon [1965]). The minimum number of edges In a simp*
Hamilton-connected graph of order n > 4 Is [\ C n + I)].

HAMILTONIAN CYCLES
221
1. If "> < li C « + I)], then a graph G with m edges contains at least one
vertex a with d^q) < 2 because, otherwise,
which is impossible.
Therefore, this graph G is not Hamilton-connected (because n > 4 and the
vertices adjacent to a cannot be the endpoints of a hamiltonian chain).
2. For W"|}C»+ I)], we shall show that there exists a Hamilton-
connected graph Gn with n vertices and m edges.
For n - 2 k, graph Gn is shown in Fig. 10.9. For n ■ 2 k - 1, graph (?„ is
shown in Fig. 10.10.
Fin. 10.10
It is left to the reader to verify the existence of hamiltonian chains of the
forms
v[a,b), v[a,xt], >«[fl.^], n[x,,yt]
ln graph Gik, and hamiltonian chains of the forms
H\a,x,\, nixt.Xj), v\xtty,\
''" graph Cafc.,.
The number of edges in G2lt equals
fc + j]-[iCfl + I)].

222 GRAPHS
The number of edges in G2k.x equals
3.2 (k - 1) + 4 .
Ml ■ 1—s - fc-3 + 2»
-3A- 1 -[2Fft-2)]-[jCn + I)].
Q.E.D.
Theorem 19 (Karaganis [1968]). If G is a connected graph of order n 2,
then its cube CJ {the composition product G • G • G) is Hamilton-connected.
Graph G3 has the same vertex set as G. Two vertices arc adjacent in G9 if,
and only if, their distance in G is < 3 (sec Chapter 4).
Consider a spanning tree // of G. We shall show by induction that II* is
Hamilton-connected. Clearly, this is true for n>2. Suppose that that is true
for ull trees of order < n und consider a tree // of order n > 2. Let a and 6
be two distinct vertices of //. We shall show that IP has a hiimiltontan chain
between a and b.
Since // is a tree, it possesses a unique chain /«[a, />] - [a, x, b) be-
between a and b. If edge [a, *,] is removed from //, two trees //„ and //„ are
formed that respectively contain a and b.
Let p[a, a'] be a hamillonian chuin in (//aK between a and a neighbour of a
in //„ (where a' is distinct from a if //„ is not a single vertex). Let n[b\ b) be a
hamillonian chain of (//*)" between b and vertex />', the neighbour of .v, in //„
(where // is distinct from b if Hb is not a single vertex).
Vertices a' and b' are separated in // by at most three edges. Thus, they are
adjacent in Ha. It follows that n[a, a'] + [a1, b'] + /«[/>', h] is the required
hamiltontan chain between a and h in II3.
Q.E.D.
Theorem 19 shows that the vertices of a tree of order n can be numbered
from 1 to n with numbers 1 and n assigned arbitrarily, such that any two ver-
vertices with consecutive numbers arc separated by at most 3 edges.'"
'" A proof for Theorem IV ilrcady uppourcd in I %0 when Sckun inn showed I hull he cube of a
connected graph hus u humiltoniun chuin. (M. Sckiinina. "On un Ordering of the Set of Venice*
of n Connected Graph." Publ. Fac. Sc. Brno, 412, IWX pp. 137-141.) Sckanina alto a»ked for
graph* whose *quurc contuinod u humiltoniun cycle. It hus recently been proved that >hc square
of ii 2-connccted graph hat a humiltonian cycle (FleUchner [1974]), thut it is Humilton-oon-
ncctod (Kobbt, Nush-Williumi). and that it it ulso punconncclcd. i.e.. if the distance between
two vcrticcn x und y is k. then there cxisu an clcmcntury chuin joining x and y with length t
0> - k, k + 1 n - 1) (Faudree, Schclp). If C is connected or order at least 4, then C
pitnconncclcd (Alavi. Williums). If G is connected or order at least 5, then the representative
graph of G or G hus a ruimiltoni in cycle (Ncbcsky). The Mjuurc (L(G)f of the rcpresentativ<
gr iph of G in vcrtcx-puncyclic. \x.. for every vertex * of (KG)I und for every integer *C < ft < "*
there exists u cycle of length k containing e (Ikrmond. Roscnstiehl).

HAMILTONIAN CYCLES
Hamiltonian cycles in planar graphs (abstract)
223
It has often been conjectured that each regular 3-conneclcd graph of degree
3 possesses a hamiltonian cycle.
The first known counter-example was the Pctcrsen graph (sec Fig. 10.11).
In fact, the Pctcrsen graph is the only graph of order < 10 without hiimil-
lonian cycles, such that the removal of uny vertex creates a graph with a
hamiltonian cycle (K. Soussclicr, in Bcrgc [1963]).
Fig. 10.11. Pctcrscn graph
Flu. 10.12. Tuttc graph
The conjecture is false even for planar graphs. Tultc [1946] constructed the
first 3-connecicd planar graph regular of degree 3 that has no humillonian
cycles (sec Fig. 10.12)
Bumcttc and JucovtC [1970] have shown that for a 3-conncctcd planar
graph without hamiltonian cycles, the smallest number of vertices is 11; an
example with 11 vertices is given in Fig. 10.13. Clearly, since this graph is bi-
bipartite with an odd number of vertices, no hamiltonian cycle exists.
M. Balinski [1961] conjectured the existence of a hamiltonian chain in nil
Planar cubic 3-conncctcd graphs. B. GrUnbaum and T. S. Motzkin [1962]
FIr. 10.13. Hcrschcl graph

224
GRAPHS
presented a counter-example by constructing a cubic, planur 3-connectq]
graph with 944 vertices in which no elementary chain could contain more tha
939 vertices (see also Brown [1961], JucoviC [1968]). The simplest example
given in Fig. 10.14, contains only 88 vertices (Zamfirescu, in Grllnbaum
[1970]).
Flu. 10.14. Zamfirescu graph
Tuttc [1956] has shown that each 4-conncctcd planar graph has a hamil-
hamiltonian cycle (sec Ore [ 1968J. for a proof).
A graph is defined to be (k)-edge-connecled if it cannot be disconnected into
two connected components, each containing a cycle, by the removal of less
thun k edges. In 1884. Tail conjectured that each simple C)-cdge-connected
planar graph regular of degree 3 has a hamiltonian cycle. The Tulle graph
(Fig. 10.12) provides a counter-example to this conjecture. Moreover, D>-
cdgc-connccicd planar graphs without hamiltonian cycles huve been con-
constructed by different authors (Hunter [1962], Lcdcbcrg [1966], Tutte[I960]).
Fig. 10.15. Another Tuttc graph

IIAM1LT0NUN CYCLES 22S
The problem is more difficult for planar E)-cdgc-conncclod graphs. The
jjrsi example (Walther [1965] of a planar E}-cdgc-conncctcd graph without
hamiltontan cycles had 114 vertices. Figures 10.15 and 10.16 present the two
simple*1 known examples with 46 vertices (Kozyrev and Grinbcrg, in Suchs
[1968]) and 44 vertices (Tultc [1972]), respectively.
Flu. 10.16. Graph of Kozyrev and Grinbcrg
Planar graphs with or without hamiltonian cycles have been constructed
by Bosak [1967]. Culik [1964], Kotzig [1962] and Sachs [1967].
EXERCISES
t. Show that the complete bipartite graph Kr,a has no hamiltonian cycles If p »* q. and
■t has [p I 2) — I disjoint hamiltonian cycles if p «■ q.
2- Show using the theorem due to Posa (Corollary 7 to Theorem 8) that if C is a graph
with n vertices, m edges and of minimum degree k, and if
m > I + max
then C has a hamiltonian cycle. Show that this is the best possible result.
(Erdos [1962])
3- Show that if C - (X. y. £) is a bipartite graph with | X\ - | Y \ - *?. | £| - m,
*>«h minimum degree Ar < 2 , and satisfies
m> I
•hen c has a hamiltonian cycle.

226 GRAPHS
4. Show that the Tuttc graph (Fig. 10.12) has no hamlltonion cycles.
Him: Use reductions to show that if the graph consisting of the triangle erf/and
interior had a hamiltonian chain, then this chain cannot have both e and/for ciulpOjn
5. Show that the graph in Fig. 10.15 has no hamiltonian cycles.
6. Show that the graph of Kozyrcv and Grinbcrg (Fig. 10.16) has no hamiltonian eye
Hint: Suppose the existence of a hamiltonian cycle n. Denote by fi the number of fi
with a contour of / edges that lie in the interior region of ;i. Denote by /!" the number r
faces with a contour of / edges that lie in the exterior region of /*• Show that
K/-2)// -n-2- I(/-2)/i".
Since /; - /.' - 0 for / 5. 8. 9. then
J/i + 6/,' + 1/i - 3A + 6/h" + 7/i".
Hence/J ■ ft mod 3, which contradicts/0' + fi ■ I.
7. If C - (X, V) is a complete symmetric graph whose arcs arc divided into two cl
V and U", show that there exist two vertices a and h and a hamiltonian circuit y. xu i
that the portion of p. from a to 6 contains only arcs of V, und the portion of ;i from 6 i
a contains only arcs of V. (H. Koynoud [I97C,)
8. In a I-graph G - (A", /'), a hamilwnhm bl-paih is defined to be two elementary path)
such that each vertex belongs to exactly one of these two paths. Let G denote thecompt*
mentary I-graph of G. Let h,,(G) denote the number of hamiltonian circuits in G, ht( )
denote the number of hamiltonian paths in C. and /tg(C') denote the number of haml
Ionian bi-puths in 6". Show that
hMG) - ho{G) + Ai(C) -t- h,(G) (mod. 2).
Show that the number of hamiltonian bi-paths in a complete anti-symmetric I-graph
is odd. (Hcrgc, |I967|).
Him: Use the proof of Theorem I. replacing the word •'arrangement" by "circular
permutation of degree n".
9. A graph G of order a is defined to be hypohamiltonian if it has no hamiltonian
cycles, but the subgraph G, obtained by removing any vertex x has a hamiltonian cycle.
Prove the following:
A) If C Is hypohamiltonian, then n > 3.
B) If C is hypohamiltonian, then lu(x) > 3 Tor each vertex x.
C) If C is hypohamiltonian. and if ,v and i arc consecutive vertices of an elementary
cycle of length « - I in G,, then x is not adjacent to both y and i.
D) If G is hypohamiltonian, then tk(x) < ■' ~ for each vertex x.
C) If G is regular of degree h, then hn - 2 m.
F) Let x, a, h. a', b' be distinct vertices of C such that x is adjacent to a and to a', b it
adjacent to b\ and arcs (a. h) and (a', b') are both contained in some hamiltonian
cycle of CK- Then, G has a hamiltonian cycle.
G) If G is hypohamiltonian, then n > 7.
(8) If 6' is hypohamiltonian, then n ft 7.
(9) If C is hypohamiltonian, then n 8.
A0) If G is hypohamiltonian, then n 9.
A1) If C is hypohamiltonian of order 10, then G is regular of degree 3.
A2) The Petcrscn graph is hypohamiltonian.
A3) Each hypohamiltonian graph of order 10 is Isomorphic to the Pcicrscn graph.
(For a complete proof, sec Her/, Duby, Viguc 11967].)

iiamii tonian rvnrs 227
q In a simple, regular graph of degree 3, consider two edges incident lo a vertex a. Show
thai >'1C P°r'ty °f 'he number of homiltonion cycles that contain these two edges docs not
d l|Pon w^ich two edges incident to a have been chosen.
II. Let C - (X. f) be a complete strongly connected graph of order n > 3; show that
each vertex lies on some circuit or length k. Tor k ■ 3.4 n.
\l, 1-d C be n complete strongly connected graph or order n > 4; show that there exist
two distinct vertices a and h such thai the subgraphs Cx-io> and 6x-it>nrc both strongly
connected.

CHAPTER II
Covering Edges with Chains
1. Eulerlan cycles
One of the oldest combinatorial problems, due to Eulcr, can be stated
follows: An eulerian chain (respectively, eulerian cycle) is defined to be
chain (respectively, cycle) that uses each edge exactly once. When does
multigraph have an eulerian chain or an eulerian cycle?
Example I. Is is possible to trace out the graph in Fig. I I.I without liftir
your pencil from the paper and without repeating any edge? After severai
attempts, the reader will find that this is impossible. However, this is pos-
possible for the graph in Fig. 11.2. which has an eulerian chain.
Fig, 11.1
FIs. 11.2
Example 2 (Euler), The city of Kttnigsberg (today known as Kaliningrad)
is divided by the Pregel River that surrounds the Island of Knciphof. There
are seven bridges in the city as shown in Fig. 11.3. Can a pedestrian traverse
Fig. 11.3
228

COVERING EDGES WITH CHAINS 229
,ch bridge exactly once? This problem puzzled the residents of KCnigsberg
ntil Euler showed in 17.16 that no solution exists.
Consider the multigraph G in Fig. 11.3 whose vertices represent the dis-
districts a, b, c, d of Kttnigsberg and whose edges represent the bridges of
Ktinigsbcrg. Clearly, the bridge problem is solved by finding an eulcrian
chai" in this graph.
Theorem 1 (Eulcr [1766]), A multigraph G has an eulerian chain if, and only
a it is connected (except for isolated vertices) and the number of vertices
Ofotid digree is 0or2.
|. Necessity. If there is an eulcrian chain n in G, then G is clearly con-
connected. Furthermore, only the two endpoints of /< (if they are distinct) can
be of odd degree. Thus, there can only be 0 or 2 vertices of odd degree.
2. Sufficiency. We shall prove by induction: // there are only two vertices
aandb whose degree is odd, then there exists an eulcrian chain that starts at a
and finishes at b; if there arc no points whose degree is odd, then the graph
possesses an eulerian cycle.
We shall assume this statement to be true for graphs with fewer than m
edges, and prove that it also holds for a graph G with m edges. To be more
definite, assume that G has two vertices a and b of odd degree.
The required chain n will be defined by a traveller who traverses the graph
from a and moves in such a way that he never uses the same edge twice. If
he reaches a vertex x b, he will have been along an odd number of the
edges incident to x, and therefore he will be able to leave x by an edge which
has not yet been used; when this is no longer possible, he must necessarily
have arrived at b. However, it is possible that not all edges have been used.
After removing the used edges, we obtain a partial graph G' whose vertices
Me all of even degree. Let Clt Ca Ck be the connected components of
G' that have at least one edge. By the induction hypothesis, these components
have eulerian cycles, Hi,na,..., /<*. Since G is connected, chain n encounters
wch of the C,. Without loss of generality, suppose that n first encounters
component d at vertex xlt then encounters component C8 at vertex x3,
*ta. Consider the chain
n[a, x,) + m + /<[*!. *j) + Ht + - + l*[xk, b).
Nearly, this chain is an eulerian chain from a to b.
Q.E.D.
The reader can now see that no solution exists for the Konigsberg Bridge
Problem (Example 2).

230 ORAPHS
Similarly, it is possible to show that the edges of a connected multigra ,
with exactly 2 q vertices of odd degree can be covered with only q chains.
Local Algorithm to Construct an Eulekian Cycle. Consider a c *.
nected multigraph G in which all vertices have even degree. The following
rules construct an eulerian cycle in G:
RuLr I. Starting from any vertex a, follow a chain without using the same
edge twice.
Rull" 2. If we arrive at a vertex x different from a after the kth step, n r
depart from vertex x along an edge that Is an isthmus of the partial Graph k
generated by the unused edges, unless x is a pendant vertex ofGk.
Rule 3. If vertex a is revisited, depart from vertex a along any unused
edge if it exists. If no unused edge exists, stop.
We shall show that the chain generated by the algorithm is eulerian.
1. It is always possible to follow the rules.
Upon arrival at a vertex x & a. there is always in Gk an edge incident >
x, because do{x) is even, and therefore dOfl(x) is odd. IP this edge is uniqu ,
it is a pendant edge in Gk and can be used. If this edge is not unique, there
is at least one other edge that is not an isthmus, since, otherwise, there are
two isthmi in Gh joining x to two distinct connected components C and
D. Note that C contains at least one vertex of odd degree, because, other-
otherwise,
0 ■ X dcSx) -1+2 mO4(C. C) mod 2.
which is impossible. Similarly, D contains ut least one vertex of odd degree.
Since Gk contains exactly two vertices of odd degree, and since one of thei i
is x, the contradiction follows.
2. If the rules are followed, the chain determined during the first k steps i
the beginning of an eulerian cycle.
Graph Gk is connected except for isolated vertices. Upon arrival at a ve
tcx x a, there arc only two vertices x and a of odd degree in Gk. Therefo
from Theorem I, GH has an eulerian chain from x to a.
Q.E.D.
We shall now apply the Euler theorem to the study of the factors of •
graph. A factor is defined to be a system of vertex-disjoint elementary cycle*
such that each vertex is contained in exactly one cycle.
Theorem 2 (Pctcrscn [1891]). IfG - (X, E) is a regular multigraph ofetX"
degree h ■ 2 k, then G has k edge-disjoint factors.

COVERING EDGES WITH CHAINS 231
I. First we shall show that G has a factor. From Theorem I, each con-
connected component of G has an eulerian cycle. By directing each edge along
the direction of travel in this eulerian cycle, we obtain a graph // - {X, V)
with
The bipartite multigraph 77 - (X, X, £) obtained by making two copies
X and .Fof X, and by taking
mulx^Xj) m mli(x,,Xj),
a regular of degree k. Thus, from Corollary 4 to Theorem E. Ch. 7). 77
contains a perfect matching £0, that corresponds to a factor in G.
2. The theorem is obviously valid for k - 1. Assume it is valid for every
regular multigraph of degree 2 k' < 2 k; we shall show that the theorem is
valid for a regular multigrnph G - (A\ £) of degree 2 k.
From part (I), we know that G has a factor £0 c £. The partial multi-
graph (X, E — Eo) of degree 2{k — I) has k — I disjoint factors Elt £a
£t.i by the induction hypothesis. Thus, G has k disjoint factors £0, £,
£*-,•
Q.E.D.
Corollary. l/G is a regular multigraph of odd degree 2 k + I such that
min mG{S,X - S) > 2 k,
s*x
S*<8
thtn G has k edge-disjoint factors.
Without loss of generality, we may assume that G is connected. Clearly,
c is a graph with an even number of vertices. Then, from Theorem A3,
C|i 8). G possesses a perfect matching. The edges not contained in the
"latching form a regular graph of degree 2 *. which from Theorem 2 can
bc decomposed into k disjoint factors.
Q.E.D.
Remark. IJablcr [1918) extended this corollary as follows: A regular graph
°fodd degree h - 2 k + I contains / edge-disjoint factors, q < k, if
min ma(S,X - S) > 2 q .
s*x
5*0

232 0RAPH3
2. Covering edges with disjoint chains
In this section we shall study the following problem: Given a simple grOpi
G ■ (X, £), cover its edges with as few as possible edge-disjoint chains ofa
certain type. What is the smallest number of such chains needed to cover £?
Example I (Kirkman). Each day, n knights meet at a round table. No
knight wants to sit next to the same neighbour twice. How many days can
they meet? The answer is the maximum number of disjoint hamiltonian
cycles in a complete graph with n vertices. Later we shall show that this
number
[n — 11
— ]•
Example 2 (Lucas). Six boys a. b, c, d, e,/and six girls a. 5, (, 3, l,J
join hands and dance in a circle. How many dances can they dance to
that no boy ever joins hands with the same girl twice and no boy ever joins
hands with another boy? The problem is solved by finding^the maximum
number of disjoint hamiltonian cycles in the complete bipartite graph Kttt,
There are three disjoint hamiltonian cycles in #«,«:
adbbceddeeffa
acbdcedfeafba
aebf cad beef da.
These cycles are found by rotating the labels of the tips of the star in Fig. I M>
Since each vertex has degree 6, clearly, these cycles cover all the edges.
Hicorem 3. The maximum number (if pairwise edge-disjoint hamiltonian
cycles in the complete graph Kn is " 7 I.
n - 1
I. First, suppose that n - 2 k + I is odd. We shall show that k - ^
disjoint hamiltonian cycles can be found.
Number the vertices 0, 1,2 2 k and place them on a circle as in Fifr
I I.S. Then, the sequence
[0.1,2.2 4.3,2* - 1,4 k + 3.*.* +2.* + 1,0]

COVEKINO BDOBS WITH CHAINS
233
•, a hamiltonian cycle. If I is added (modulo 2 k) to the index of each vertex
. 0, i.e. if the system of solid lines in Fig. 11.5 is rotated around
point 0. then a new hamiltonian cycle is obtained. This cycle is disjoint
from <ne preceding one because the sum of two successive vertices is 2 or 3
,n the first cycle and 4 or 5 in the second cycle. Thus k - 1 rotations around
point 0 can be made, and these rotations produce k disjoint hamiltonian
cycles.
2. Suppose that n - 2 k + 2 is even. Number the vertices of the graph
0.1.2, ...,2 k, a, and place them on a circle as shown in Fig. 11.5. with
vertex a placed on another phinc above the centre of the circle. Similarly.
k disjoint hamiltonian cycles can be produced by rotating the solid lines in
Fig. 11.5 around point 0. Thus Kn contains k - [^-y -J disjoint hamil-
hamiltonian cycles.
Q.E.D.
Corollary I. If n is odd, the edges of graph Kn can be covered by ——-
disjoint hamiltonian cycles.
Since the degree of each vertex is n - I - 2 k, the k disjoint hamiltonian
cycles cover all the edges.
Q.E.D.
Corollary 2. Ifn is even, the edges of graph Kn can be covered by a perfect
hatching and \ ^-^- 1 disjoint hamiltonian cycles.

234 ORAPHS
If the k hamiltonian cycles are removed, the remaining partial graph,.
regular or degree (/» - I) - 2 k - I.
QE.D.
Corollary 3. Ifnis even, the edges of graph KH can be covered by ^ disjo /
hamiltonian chains.
The edges of Kn< t can be covered by disjoint Hamiltonian cycles as shown
in Corollary I. If we remove one vertex, these cycles correspond to "disjoi i
hamiltonian chains of Kn, 2
QE.D.
Corollaries I and 3 are also consequences of the following general theorem;
Theorem 4 (Loviisz [ 1968)). The edges of a simple graph G with n vertices c n
always be covered by v elementary chains and cycles that are pairwlse edg
disjoint, or less.
1. We may assume that the graph G hits no isolated vertices and no iso-
isolated edges. Otherwise, we could prove the theorem for the graph obtained
by removing isoluted vertices and edges.
2. Let m denote the number of edges in graph G. If 2m — n < 0, then
Since </<,(*,) > I for all /, we have d,,(x,) - I for ull /. This contradicts the
hypothesis that G has no isolated edges. Thus 2/w - n > 0.
3. We shall assume that the theorem is valid for each graph G of order
n' with
[y]-m
and 2 m' — «' < 2 m — n
and we shall prove the theorem by induction on 2 nt — n.
4. First, suppose that there is a vertex of even degree. Let .v be such that:
rc(x)" {ax, a2 ak,bi%b2 b,},
<lc(ai) even , dc(bj) odd , k > I .
Let G' be the graph obtained from G by removing edges [a,, x] for / " '
2 k. Graph G' satisfies 2m' - n' < 2m — n, and contains a minimui»
covering M' - M,/^. •••} of less than F" 1 disjoint elementary chains an
cycles (by the induction hypothesis). ^ J

COVERINO EDOM WITH CHAINS
235
Since vertices a, and b, have odd degree in G\ each of these vertices is the
ndpoiut of an elementary path of M' of length > 0. Since the covering M'
. minimum, each of these vertices is the endpoint of exactly one path of
5, Thus vertex a, is the endpoint of a chain n'[a,,z,] e M'. Suppose that
i.zil- ^ yl denote the next to the last vertex encountered in the
jf;[
partial chain /('[a,, x]. Since vertex yl is necessarily a b,, it follows that y\
is the endpoint of a chain p'[y}, zf] e A/', If x e n'[y\, r,1), define similarly
8 vertex >•?, etc.
In this way, a finite sequence
is determined for each / < k. It is easy to see that >f «■ y) implies p - q
and (' " / Furthermore, the set Y of all yf (for all / and all p) satisfies
C
FIb. 11.6
6- Now, we shall show that a covering (/*,) of G is obtained from covering
A/' of C in the following way:
Ir ti e A/' has an endpoint in Y, let
Hr "

236 ORAPKS
Thus, n'r is an elementary chain or an elementary cycle or G.
If n',[yf, z]] e M' has in Y only one endpoint, say yf, let
- {ri + Of. x] } - { Or* '. *] } otherwise.
Thus, n> is an elementary chain of G.
ile A/' has two endpoints yf and j1? in Y, let
0. - *; + or. *] + 05. *] if ^
- M + Of.*] + OJ.x]} - {Of*'.x].Or!.*]} otherwise.
Thus, in the first case, ft, is an elementary cycle of G, and in the second case,
Hi is an elementary chain of G.
To show that the set M of all the chains n, forms a covering of G, note
first that
— an edge of G that is not of the form [yf, x) is also an edge of G' and i|
covered by a single chain of A/,
— an edge of the form [yf, x] is covered by the only chain of M that comes
from the chain p.', (or n',) that has yf as un cndpoint.
7. Now, suppose that each vertex of G is of odd degree; consequently,
n is even.
Replace G by a graph Gx obtained by taking a vertex x of odd degree * 3'
and adding a vertex al in the middle of some edge incident to x, say [x,z].
Since graph G\ (obtained from graph Gx as in Part 4) satisfies
2 m\ - n\ - 2 m - (n + I) < 2 m - n ,
Gx can be covered by I y I ■ j disjoint chains. Each vertex of Ci that i<
distinct from «i is the cndpoint of a covering chain. Hence, a, is not the end-
point of a covering chain. Thus, the - covering chains of Gx ulso cover •
Q.E.D.
Corollary. Let G - (X. E) be a simple graph of order n with only vertic I
of odd degree; then its edges can be covered by =-«fe* disjoint elementary chain
so that each vertex of G is the endpoint of exactly one of these chains.
Since 2 */0(a-) - 2 m is even, then n is even. Therefore, from Theorem 3, ■

COVERINO EDOBS WITH CHAINS 237
he edgcs of G can be covered by '- disjoint chains and cycles. Furthermore,
aCh vertex is necessarily the endpoint of at least one covering chain, because
i($ degree is odd.
Q.E.D.
We shall now state three conjectures related to Theorem 4.
Conjecture 1 (Haj6s). If all the n vertices have even degree, the edges can
be covered byl'l] edge-disjoint elementary cycles, or less.
Conjecture 2 (Gallai). The edges of a connected graph of order n can be
(ocered by [ —=— 1 edge-disjoint elementary chains.
C<wi/«:ri/r*3(Nash-Williams). Forn > \5, if the number of edges is divisible
by i and if each vertex has even degree > -j- • then the edges can be covered
by edge-disjoint triangles.
Theorem 5 (ErdOs, Goodman, P6sa [1966)). Let G be a simple graph of
order n; then the edges of G can be covered by [n*/4] edge-disjoint cliques, each
of cardinality < 3.
1. The theorem is true Tor n - 2. Assume that the theorem is valid Tor
all graphs with n — 1 vertices, we shall show that it is also valid for a graph
C of order n > 2.
First, we shall show that
- Ik, then
l.then
2- If G contains a vertex x0 with degree dc(x0) <l = 1, the subgraph Co
Stained by removing x0 can be covered by I [cliques of cardinality
^ 3 that arc pairwise disjoint. Thus, the minimum number of disjoint cliques
ht>« cover the edges of G is less than or equal to

238 OKAPH3
3. If each vertex x satisfies do(x) >N1< the number k « minX(X
can be written as
k-\'z\ + r • r>0
where
Let a:, be a vertex of degree A > 2 r. We shall show that the subgraph
generated by F^xJ contains a matching of cardinality r.
In Tact, if this subgraph contains a matching or only r - 1 edges
[jWal. [>'3')'i] [.ftr-s..Var-a). vertex ;'ar., is not adjacent «°
which contradicts that k equuls the minimum degree.
4. Let ro(x,) - {>, ,ya yk), and let C, be the partial subgraph obtained
by removing xx and the r edges of the matching [yitya] [^'ar-i.^ar] &*"
fined above. The edges of graph C can be covered by:
— r triangles [x,,ylt ya, x,] [x,, >ar.,,)>„, x,].
— k - 2 r isolated edges [x,, y2T-i] [xltyh],
— the I —2— I disjoint cliques that cover Gx.
Thus the minimum number of disjoint cliques that cover G is less than or
equal to
Q.E.D.
Remark. This theorem is the best possible, i.e., the edges of some gr»P"

COVF.K1NO EDGES WITH CHAINS 239
[n2l
-t I cliques of cardinality < 3.
por an even n - 2 k, let Cn be the complete bipartite graph Kk_h. The
minimum number of cliques needed to cover all the edges is
for an odd n - 2 k + 1. let Gn be the complete bipartite graph
The minimum number or cliques needed to cover all the edges is
k(k + 1) -
J. Counting culerian circuits
Bk+ IJ 1 rM2l
4 ~4"I 4 J "
Let G - (X, U) be a (directed) graph. A circuit of G is defined to be
tulerian if it uses each arc exactly once. In this section, we shall study the
number of distinct culcrian circuits in G.
Graph G is defined to be pseudo-symmetric if the number of arcs entering
vertex x equals the number of arcs leaving vertex x, for all vertices x, i.e.
dHx)mdo(x) (xeX).
This terminology is consistent because each symmetric graph is also pseudo-
symmetric.
'I hcorcm 6. A graph possesses an valerian circuit ff, and only if. it is con-
nected {except for isolated vertices) and is pseudo-symmetric.
The proof is exactly the same as the proof of Theorem 1.
Example. What is the longest circular sequence that can be formed from
the digits 0 and 1 so that no A-tuple of k consecutive digits occurs twice?
Since 2" distinct k-tuples can be formed from the numbers 0 and 1, the se-
sequence cannot have more than 2" entries. Using Theorem 6, we shall show
that there does exist such a sequence with 2" entries.1
Consider a graph G whose vertices represent the different (k - l)-tuplcs
ofOitnd 1 with arcs from vertex (a,, aa,.... a*.,) to vertex (aa,a3, .... «*-i,
°) and to vertex (aa, a3,..., a*.,. 1). Since graph G is pseudo-symmetric, it
Possesses an eulerian circuit. If (a,.aa aid) « the first vertex of this
c'rcuit, (a3, «3 ak) the second vertex, (a3, a, a*.,,) the third vertex,
ctc. the required sequence will be a,, aa, a3, a,
For k - 4, the graph in Fig. 11.7 provides several circular sequences of
2' ■ 16 entries. For example,
'"This problem occur* In telecommunications when it l» nceetmy (o determine (he current
Dil of • labelled cylinder by reading from the cylinder a nequence of only Ac of lit labels.

240
ORAPKS
fl-0000
abcdefgh/jk/m/tpq
abcdk/jcfgh/mnpq
abcdk/pgh/mwjcfq
abcfgh/jcdk//?mpq
abh/jk/m/tpgcclcfq
abh/jedk/m/tpgcfq
abhijcfgcdk/m/tpq
abh/pgcdk/mnjefq
0000101001101111
0000101101001111
0000101100111101
0000100110101111
0000110111100101
0000110101111001
0000110100101111
00001100101II101
(Circular sequences obtained by permuting / and Imn have been omitted.)
There are 16 different solutions.
The problem of counting the culcriun circuits is closely related to the prob-
problem of counting the spanning urborescenccs, sec Theorem B1, Ch. 3).
rhcorcm 7. Let G be a connected pseudo-symmetric graph, and lei xx be a
vertex ofG. Then G has a partial subgraph which is an arborescence rooted
at Xi. Moreover, this arborescence H can be constructed by travelling through
an eulerian circuit starling at xt and placing in II the first arc used to ent r
each vertex.
From Theorem A3, Ch. 3), // is an arborescence rooted at xx since:
1. each vertex distinct from xt is the terminal endpoint or exactly one
arc of H,
2. *i is not the terminal endpoint of any arc of //,
3. // does not contain any circuits, because if a path formed from the
arcs of // goes from x to y, then vertex x has been visited before vertex
y, and therefore no paths exist in // from ;' to x.
Q.E.D.
Theorem 8 (Aardcnne-Ehrenfest, dc Bruijn {1951]). In a connected pseud*
symmetric graph G, let Ax denote the number of distinct arborescences root

COVERINO EDGES WITH CHAINS 241
- Xl that are partial graphs, and let rk denote the outer demi-degree (or inner
• ^.degree) of vertex xk. There are exactly
dl fl (rk - 1)!
k-l
distinct eiderian circuits.
Note that two culcrian circuits are not considered as distinct if one can
be obtained from the other by a circular permutation of the arcs.
Consider an arborescence // rooted at xt that is a partial graph of G. We
shall show that there arc exactly
n
no*- !>!
k' i
distinct culcrian circuits that produce // as in Theorem 7. Number the arcs
entering xk from I to rk and let
t»~(Xk) - { Mt(l).UiB)....,.
Suppose that uk(rk) is the arc of // in w(xk), for each k I. Suppose that
arc «i(l) is fixed. Then there are exactly
possible numbcrings. Each numbering corresponds to exactly one circuit
as follows: Starting from *„ travel in reverse through all the arcs of the
graph by choosing at vertex xk the unused arc of a»"(.vfc) with the smallest
possible number.
This procedure defines a circuit because the route can terminate only at
*i (when any other vertex is encountered there is always an exit arc because
the graph is pseudo-symmetric). We shull show that this circuit is culcrian.
If it were not culcrian, then there is an arc uk{j) that is not used by the
circuit. Since rk > j, arc uK(rk) has not been used. Hence, arc «„(/•,,), incident
'"to the initial end point xp of arc uk(rk\ has not been used. etc. Since each of
Jhcsc unused arcs belongs to //. the process eventually arrives at xx. But this
ls a contradiction, since the procedure that generated the circuit stopped at xt,
aid consequently, all arcs incident to xl have been used.
Thus each numbering corresponds uniquely to an culcrian circuit such
"at the first arc of the circuit to enter a vertex is present in the arborescence
'• Since the number of distinct arborcsccnccs // is Alt and is known by
T|icorem B1, Ch. 3), the proof is complete.
Q.E.D.

242 ORAPHS
Corollnry 1. In a pseudo-symmetric graph, the number of arboresctntt
rooted at xk that are partial graphs is independent of which vertex xk is chosen
Thcorein 8 can also be stated Tor vertex xk instead or Tor vertex .v, Since
the number of eulerian circuits docs not change, At ■ Ak.
Q-E.D.
Corollary 2. In a graph G with m arcs and with order n that satisfies w > 2 u
the number ofeulvnan circuits is even.
If G is not connected and pseudo-symmetric, the result is obvious
) < 2 for each vertex .v, then
m - £ dt(x) < 2 n,
X
which is impossible. Udc(x) > 3 for x - xk, then {rk - I)! is even and the
formula in Theorem 8 shows that the number of distinct eulerian circuits
is even.
Q.E.D.
Note that this result is the best possible: each graph shown in Fig. 11.8
satisfies m - 2 n and contains exactly one eulerian circuit.
COOOO
Mr. 11.8
Corollary 2 can be extended by using a fundamental theorem •
algebra, due to Amitsur and Lcvitzki [1950], that was reformulated in more
combinatorial terms by M. P. Schutzcnbcrger A958] as follows.
Let C be a graph of order /i whose arcs are numbered I, 2,..., m. To each
eulerian circuit (\,.h,J3, •■■Jn) beginning with arc I, there corresponds >
permutation
il 2 ... M\
H h ■■■ U

COVRR1NO EDOES WITH CHAINS 243
An culcrinn circuit fi is defined to be odd (respectively, even) if the corres-
corresponding permutation is odd (respectively, even). We shall show that if
„, > 2 n, the number or even culcrian circuits equals the number or odd
eiilcrinn circuits. (Note that if the arcs arc numbered differently, the equality
i$ preserved.)
Ultima. Let n — (jltja, ...,jm) be a permutation of\, 2,..., m and let
nk m (Jl'Jlt •■■ i Jk-l>Jk + ltJk + 3> ■■■• Jm)
be a permutation of degree m — 2 obtained from n by the elimination of two
consecutive indices Jk,jk + l; then n and nk have the same parity if and only if
one of the following conditions holds:
A) Jk<J* + \ and J
B) Jk>J\n and Jt+Jk*i even.
Permutation n - (jiJa Jm) becomes (/,,/, A-i.A + i Jn,Jk)
by a product or m - k transpositions of two consecutive terms. Similarly,
this new permutation becomes (jiJa Ju-\Ju»2,ik*t JaJk.Jmi)
by a product of m - k transpositions. Thus n becomes (Ji,Ja JmJkt
Am) by a product of 2(/n - k) transpositions.
Two permutations arc of the same parity if one can be obtained from the
other by an even number of transpositions: thus, permutations n and
O'i.,/a. -,jm,JkJkn) have the same parity. By removing./* and /„„, from
this latter sequence, the number of inversions is decreased by
"Vt < A»i, this number equals
(,„ -Jk - |) + („, _ A+1) . 2wi - I - (A +A+i) ■
"a > j\*i, this number equals
(m - JO + (/« - jk+,) - 2 »i - (Jk + Jk* i> •
Since permutations n and nk have the same parity if, and only if, this number
ls even, the lemma follows.
Q.E.D.
Theorem 9 (Schlitzenbcrgcr [1958]). In a graph G - (X. V) of order n with
'" arcs that satisfies m > 2 n, the number of even eulerian circuits equals the
"umber of odd eulerian circuits.
'• We may assume that m ■ 2 « + 1. Otherwise we have m - 2 n - A: > I.

244 ORAPHS
Then, construct a graph C from G by adding A- — I new vertices and J^
replacing one arc (.v. y) e V by an elementary path from x to y that pa ■
through each of these A- - I new vertices. Graph C satisfies m' - 2;'
(;n + A- - I) - 2(» + A1 - I) - m - 2 n - k + I - I, hence, C has a$
many even culcrian circuits as it has odd euleriun cycles, and the same result
follows for G.
2. Note that ihc theorem is true for a graph of order I or 2. We shall
assume that it is valid for all graphs of order < ;i. and we shall show that
it is also valid for a graph G of order ;i > 2, with m - 2 it + I.
3. We may assume that G is connected and satisfies
^M-JcW (xeX).
Otherwise, from Theorem 6, the number of eulerian circuits equals 0, and
the theorem is proved.
4. We may assume that C is a I-graph. Otherwise, there are two arcs
/, and /'a with the same initial endpoint and the same terminal endpoint.
With the eulerian circuit corresponding to a permutation n, we can associate
the eulerian circuit /<' corresponding to the permutation n' obtained from x
by transposing i\ and !a. Since fi and n' have different parities, the theorem
follows.
5. If G contains a vertex x with </J(x) - </J(x) - 2 and with a loop at x,
the theorem can be proved as follows: The three arcs incident to x are of
the form (xo,jc) -/0, (x,x)-/,, (x,x,)-/2.
Consider the graph C obtained from G by removing x and all arcs inci-
incident to x, and by adding an arc i0 - (x0, xa). Graph C has order ri ■
n - I and satisfies m' - In' - {m - 2) - 2(» -1) - m - 2 n - I. Thus,
from part 2, C has as many even eulerian circuits as it has odd eulerian
circuits. To each of the corresponding permutations, add the entries it,
after the entry /0. In this way, each culcrian circuit of G is obtained. If. for
example, /t < ia and /| + ia is even, two circuits of the same parity in C'
correspond to two circuits with the same parity in G. Thus, the theorem is
valid for G.
6. If G contains a vertex x with d$(x) - d£(x) • I. the theorem can b«
proved as follows: By making the transformation shown in Fig. H-9< a
graph G, is obtained for each arc /ecu'(*„)■
From Part 5, each graph G, has the required property.
To each eulerian circuit in G,, there corresponds an eulerian circuit in 0-

COVERING EDOE8 WITH CHAINS
245
To each culerian in G there corresponds an culerian circuii in only one of the
C|. Hence G" hus (he required property.
Fl«. 11.10.
7. In ull the other cases, we shall show that (he outer demi-degrec of each
Vcncx in G cquuls 2. except for one vertex jc, with da (xt) - 3. From Part 6,
*c may assume that d£ (.v) > 2 for all x, and consequently
2n + I - in - T do(x) > 2« .

ORAPHS
a.
Flu. 11.il-
Thus, da (.v) - 2 Tor all x at, . Since G has order n > 2, it contains the
configuration shown in Fig. 11.10.
8. The configuration in Fig. 11.10 can be altered in Tour ways as shown
in Fig. 11.! I. These alterations yield the graphs C,. Ga, G7, G8, each of order
n and with m - 2n - 1.
Let n(G), no(G) and nt(G) respectively denote the number of culerian cir-
circuits, the number of even culerian circuits and the number of odd eulerian
circuits in graph G. Then
n(G) -
n(G2) - n(Gt) -
From Part 6,
Furthermore, from Part 5,
Thus,
no(G2) - nQ(Gt) - no(G7)-

COVERINO COOES WITH CHAINS 247
Thus, G has the required property.
Q.E.D.
EXERCISES
I ^ c - IX, £) be a simple connected graph regular of degree 4. and If (£', f) is a
partition of the edges into 2 factors: show that there exists an eulerian cycle in C th*t
alternately passes through edges of E' >nd F'. (Kotzig [1956))
2. Use Theorem (l5,Ch.3), and Theorem F, Ch. 11), to give a necessity and sufficient con-
condition th*t > seml-functlonal I-graph possesses a hamiltonian circuit.
3. A simple graph L(C) is defined to be the lint-graph of C - (X, E) if each vertex of
L(G) represents an edge of C and if two vertices are adjacent in L(C) if they represent
■djiccnt edges of C. (These graphs are studied In Chapter 17, Section 4.) Show that if C
poucsses an eulerian cycle, then L(C) possesses also an eulerian cycle. Show that the
converse is not true.
Hint: Consider a graph G with 4 vertices a, b, c, d and 3 edges ab, be, bd.
(Chartrand [1964))
4. Show that If G possesses an eulerian cycle, then L(C) possesses a hamlltonian cycle.
Show that the converse Is not true.
mm: Consider the graph C with 4 vertices a, b, c. d and 6 edges ab. ac, ad, be. bd, ed.
(Chartrand [I964J)
5. Show that if C possesses a hamiltonian cycle, then HO possesses a hamlltonian
cycle. Show that the converse is not true.
Hint: Consider the graph C with 5 vertices, a, b, c, d, t and 6 edges ab, ac, ad, ib. tc, ed.
(Chartrand [1964))
6. Show thai if C b a regular multigraph of order n and odd degree h ■ 2k + I with
"> n + I, then C has a factor.
Hint: Use the proof of Theorem 2, and Corollary I to Theorem E, Ch. 7).

CHAPTER 12
Chromatic Index
1. Edge colourings
The chromatic index q(G) of a graph G is defined to be the smallest number
of colours needed to colour the edges of G so that no two adjacent edges have
the same colour. A q-colourlng o/tlie edges is defined to be a partition of the
edge set into q subsets that arc matchings.
Clearly.
max dc{x).
In this chapter, we may assume without loss of generality that G is a multi-
graph without loops.
Example 1. Scheduling an oral examination. At the end of the academic
year each student must be examined orally by each of his professors. How can
the examinations be scheduled so that they end as soon as possible?
Let X be the set of students, and let Y be the set of professors. Form a
bipartite multigraph G - (X, Y. E) in which a e X is joined to be Y by k
edges if, and only if, student a must be examined by professor b exactly k
times.
If the edges of this graph are coloured so that no two adjacent edges have
the same colour, each colour can correspond with one examination period.
Hence all examinations can be completed in q(G) time periods, and not less.
Example 2. The Lucas schoolgirls problem. Each day the 2p schoolgirl*
of a boarding house take a walk in p rows of two; can they take 2p • '
walks consecutively without any two girls walking together more than once?
This is possible, if and only if the chromatic index of the complete graph Kit
with 2p vertices is equal to 2p — I, each colour representing a walk.
Example 3. latin square. A latin squire is an n x n matrix with entries
1, 2,.... n such that no entry appears twice in the same row and no entry
appears twice in the same column.
Let k < n; in an n x n tablcuu. place only entries 1. 2 k, so that n°
entry appears twice in the same row or in the same column. Is it possible to
248

CHROMATIC INDEX
249
place the entries k + 1, k + 2,.... n in the empty positions so that a latin
Jquare is formed ? Represent the rows of the matrix by vertices ax, a , an,
and the columns by vertices bltba,.... bn; join a, and *, if and only if the
position in row /, column./ is empty. The latin square can be completed if,
and only if, the edges of this bipartite graph can be coloured with n —k
colours k + I, k + 2,.... w, each edge [a,, b,] coloured with a corresponding
to an entry a at the intersection of row / and column/
Theorem 1. The chromatic index of a simple complete graph G of order n
(and maximum degree h - n — I) is
. | - h Ifn Is even
9V M - It + 1 ifn is odd.
Case I: n is even. Number the vertices 0, I, 2 n — I and place the
vertices as shown in Fig. 12.1.
I
Fl«. 12.1
Let the first perfect matching (i.e., the edges of the first colour) be
[O.!].[2.m-I].[3.m-2]
These edges are the dark edges in Fig. 12.1.
By adding 1 modulo n to the index of each vertex except vertex 0 in the
above matching, we obtain another perfect matching, which corresponds to a
rotation around 0 of the dark edges in the figure. This operation can be per-
performed " — I times, and each time a new colour is assigned to the resulting
"latching. No edge is coloured twice, because each time the coloured edges
•orm a different angle with ihe horizontal. However, each edge of G has been
coloured, and therefore q(G) - n - 1.

250 ORAPHi
Case 2: n is odd. Consider the graph C formed from C by adding a
x0 that is joined to each vertex of G. From Case I above, we have
q(C) - (n + I) - I - n.
Thus the edge of G can be coloured with n colours. The edges of G cannot
be coloured with n — 1 colours, because then each colour would correspond
to a perfect matching, and a graph with odd order has no perfect matching.
Thus
q(G) m n m h + I .
QE.D.
Theorem 2. The chromatic index of a bipartite multigraph C - (X, Y,E)
with maximum degree h is
q(G) - h.
From Corollary 4 to Theorem E, Ch. 7), graph C contains a matching £,
that saturates every vertex of degree h. Colour Ex with the first colour.
Consider the bipartite multigraph (X. Y.E— Ex), whose maximum degree
is k — 1. This graph contains a matching £a that saturates every vertex of
degree h — I. Colour £a with the second colour. By repeating this operation,
the edges of G will be coloured with h colours.
Q.E.D.
Application to latin squaki-s. The theory of latin squares, which was
founded by Euler, can use Theorem 2 in a very interesting way: a latin
square of order n is formed when the entries 1,2 n are distributed into the
ri1 positions of an n x n square so that no entry is repeated in any row and no
entry is repeated in any column. We shall show:
Let T he a p x a rectangle whose positions contain the numbers 1. 2, ..•. "
such that no number is repealed in any row and no number is repealed in any
column. Let m{k) denote the number of limes that k appears in T. A neces-
necessary and sufficient condition that n — p rows ami n — q columns can be added
to form a latin square is that
m(k) > p + q - n (k - 1, 2,...,«).
Necessity. Let m'(k) denote the number of rows of T in which k does not
nppear. If a latin squnre can be formed, then m'(k) < n - q. Hence
m(k) - p — m'(k) > /> - n + q.

CHROMATIC INDEX 231
Sufficiency. Let
A*-{1.2 p), AT - { 1.2 «},
form the bipartite graph G ■ (P, N, E) in which / e P and k e N are joined
by an edge if number k does not appear in the /-th row of T. Then,
dc(i) -n-q (teP)
da{k) m m'(k) < n - q {k e N).
From Theorem 2, the edges of G can be coloured with n — q colours.
Complete T by adding n — q columns; the first additional column will be
defined by the endpoints in N of the edges of the first colour, the second
column by the endpoints in N of the edges of the second colour, etc. In this
way. we obtain n p x n rectangle T, thnt satisfies also the hypothesis of the
ihcorem because ifl(y) ■■ p - (p + n) - n. Then n — p rows can now be
added to complete T, and this yields a latin square.
Q.E.D.
Consider the problem of colouring the edges of n bipartite graph with
colours O|, oca a, such that there are exactly m, edges with colour a,. The
following theorem gives conditions for the existence of such a colouring.
Theorem 3 (Folkman. Fulkcrson [1966]). Let G - (AT, Y, E) be a bipartite
multigraph with maximum degree < q. Consider a sequence
with
mi, - »i - I £ |.
1-1
Form the conjugate sequence (mf, m*,...) where m* denotes the number of
»'i that are > j (see Chapter 6). If there exists a colouring of the edges in q
colours O|. aa. .... a, with exactly mt edges with colour a,.f>r all i, then
mo{A,B)> T m*, (AcX,B<=Y).
J>\X-Aj*-\r-B\
This necessary condition is also sufficient for the case:
m, - nij ■ ••• - mk > ws + | - /nA4.2 ■ "■ ■ wf.
Necessity. If there arc m, edges with colour a,, then the number of edges
w>th colour a, thut join A and B is
, B) m /,,, _ nl£X - A,Y)- /«,(*, Y - B) + m^X - A.Y - B) >
>m,-\X-A\-\Y-B\.

252
Hence
ma(A,B)m £ w,M.B)>
ORAPHJ
(The last equulity can be verified from the Ferrers diagram, see Chapter
6. Section I.)
Sufficiency {for the special case). Consider such a sequence (m,). See
Fig. 12.2. Let
W ■
(■1
«
Ml
/,' . fc; h" - 9 - k; A' + A" - 9 .
We shall show first that C is the union of two disjoint partial multigraphs
C - {X, Y, £') and C" - (AT, K. £') with
| £' | - m', | E" | - W ,
max dc.(z) < /)', max dQ-(z) < /)".
From the lemma to Theorem G, Ch. 7), it suffices to show that for all A, B,
A) mo(A,B)>m' -W\X-A\-W \Y-B\,
B) mG(A,B)->m" -h*\X- A\-h"\Y- B\.
in'
in"
h"
hi
•

CHROMATIC INDEX 253
Inequality (I) can be easily verified from the Ferrers diagram (Fig. 12.2).
If
— <| - A\ + \Y- B\,
then
J>\
If
I m*mh'K-\X-A\-\Y-B\\m
x-A\*\r-a\ \/i /
mm' -h'\X-A\-W\Y-B\.
~>\X- A\ + \Y- B\,
then
T mJ>m'-h'\X.-A\-h'\Y-B\.
j>\x-A\-\v-a\
In both cases the inequality of the hypothesis yields inequality A).
To verify inequality B), note that if
~ >\X-A\ + \Y-B\,
then
- m" - te | x - a | - /r i y - fl |.
and if
W
then
0 > h" (£ - | X - A | - | Y - B
Thus inequality B) is satisfied in both cases.
From Theorem G. Ch. 7). multigrnph C (with m' edges and muximum
.„<
< A') contains a mutching £,' with X7 - w, edges that saiunitcs
each vertex of maximum degree. Colour the edges of E[ wiih the first colour.

254 ORAPHS
The partial multigraph generated by £' - E[ has maximum degree < h -
From Theorem G, Ch. 7), this graph contains n matching £J
-r, r-1 ™ mi edges. Colour the edges of E'% with the second colour, etc.
Thus C can be decomposed into h' - k mulchings of mt edges. Similarly,
G" can be decomposed into h" - q — k match ings of/Mk+l edges. Thus C
can be coloured with q colours so thnt m, edges are of colour a, for all /.
QE.D.
2. The Vlzlng theorem and related results
This section presents some bounds for the chromatic index of a multigraph
without loops. The following very simple theorem gives n lower bound.
Theorem 4. Let G be a multigraph without loops, with m edges, with maxi-
maximum degree h, and let t be the cardinality of a maximum matching. Then,
q(G) > max
{*■[?]')■
Consider a colouring of the edges with q ■ q{G) colours a t, aa,...,«,, and
let £, denote the set of edges with colour a,. We have
m - | £, | + | E21 + - + | £J < qt.
Hence, q > - , and q > [ - I . The theorem follows.
Q.E.D.
Uncoloured Edge I emma. Let G be a multigraph without loops and let
q(G) - q + I. Suppose that a set C - { a,,aa, .... a,} of q colours has been
used to colour all the edges ofG except one edge [a, b]0, and let Cx denote the
set of colours used for the edges incident to vertex x. Then
C. n Cb | - do(a) + do(b) -q-2
C. - Cb | - q - do(b) + I
C» - C. I - q - da(a) + 1 .
No colour of C can be missing from both Ca and C*, since then [a, t>\t
could be coloured with this missing colour. Thus C ~ CaKJ C«, and
q - | C\ - | C. u Cb\ - | C. n C» | + | C. - Cb\ + \ Cb - C. \.

CHROMATIC INDEX 2SS
Furthermore,
do(a) - I - I C.| - | C.n C»| + | C. - C»|,
</„(*)- 1 -IQI-IC. nCb | + \Cb-C. |.
py elimination, these three equalities yield:
+ do(b) - I - <y,
- I),
I Cb - C.\ - v -(da(b)- I).
Q.E.D.
Ihcorcm 5.//G consists of a cycle [xlt xa,.... x*, Xi], with possibly more
than one edge between two consecutive vertices, and if G has m edges and
degree h. then
h ifn is even
fl(O- t r 2 m 1*1
max | h, [ ^-—y \ ) ifn is odd.
If ;i is even, the proof is evident because then G is bipartite and Theorem 2
applies. Thus we may assume that n - 2 k + 1 odd.
Note from Theorem 4 that
ll remains to show that
*O<max{*t [£]*).
This is true for a multigraph with 2 it + I edges, because </(G) - 3 for a
cycle G without multiple edges. Therefore, we may assume that m > 2 k + I.
Suppose that the theorem is valid for all multigraphs with less than m edges,
and consider a multigruph G with »i > 2 k + I edges.
'■ If we remove from G an edge [a, fe]0 so that a and b remain adjacent,
'he resulting graph G' is of the same type but with m' - m — I edges. Thus,
from the induction hypothesis, the edges of G' can be coloured with a set C
°f<7 colours, where
For a e C, let Ea denote the set of edges with colour a.

256 oraphs
2. We shall assume that q(G) - q + I and we shall show that this leads tj
contradiction.
There is in C a colour y with | £, | < k since, otherwise,
and
m
k k
which contradicts q > J — | .
Two cases must be considered: yeC»-Ca (or yeCa-Cb) and
yeCanCb. From the Uncoloured Edge Lemma, C- CavCb, and no
other cases are possible.
Case I: y e C» - Ca. From the Uncoloured Edge Lemma,
I C. - Q | - q - dc(b) + I > h - dc(b) + I > I .
Thus there exists a colour beC, - Cb, and clearly a »* y. Let G(a, y) denote
the partial graph of G generated by the edges with colour a or y. The con-
connected component of G(x, y) that contains b is an elementary chain with end-
point b (because a i Cb) and does not contain vertex a (because | E, \ < k).
By interchanging the colours y and a along this chain, colour y is no longer
incident to vertices a and b and, therefore, [a, A]o can be coloured with y,
which is a contradiction.
Case 2: yeCanCb. From the Uncoloured Edge Lemma, there is a colour
a Ca- C and a colour /? € C» - Ca. Clearly, y »* a, /?.
Consider the connected component of graph G(fi, y) that contains a. This
component is an open chain naf[a, x) that contains b if y joins vertices a and
b, and does not contain b if y does not join vertices a and b.
Since this chain begins with an edge with colour y, by interchanging y and
fl along this chain a new colouring with | Er \ < k is obtained.
Furthermore, in this new colouring, yeCb— Ca, which is Case I.
Q.E.D.
We shall now give an upper bound for the chromatic index. The strongest
theorem of this type is due to Vizing [1964], and was rediscovered by R. P-
Gupta [1966]. It is given in a slightly stronger form below:
Theorem 6. Let G be a muhlgraph without hops. Let [a, b]0 be an edge ofC,
and let G' - G - [a, b]0. If the edges of G' can be coloured with q colours,
where q > </0(a). q > d^b). and if

CHROMATIC INDEX 257
x e rc.(a) * dc.(x) + mc.(a, x) < q ,
the edges of G can be coloured with / colours.
Suppose that G cannot be coloured with q colours. Then.
q(G')-q, q(G)- q+ I.
1. Let Cx denote the set of colours used for the edges of G' incident to a
vertex x6 ra(a). Then.
| C - C, | - q - dc.(x) > mc(a, x).
Therefore, there exists a mapping g(e) of the edge set o>c(a) into the set C of
colours such that
(i) e,-[a,x), * g(e,)tCx,
(ii) ek"[a,x]k, ej-[a,x)j, k j m- g(ek) + g(e,).
2. We shall define a sequence y0, yi, y3l... of distinct colours in Ca as
follows:
Let y0 be a colour in Ca — Ct; such a colour y0 exists because, from the
Uncoloured Edge Lemma,
\C.-Ck\-q- do(b) + 1 > I .
Let t'i - [a, Xi]i be the edge with colour y0 that is incident to vertex a.
Lct yi ■ ?(^i)- Clearly, ft y0. since yi ^ C,, and y0 e C,. We have
yi c C because, otherwise, yi ^ Ca, yi ■ jK^i) ^ CAi, and we can recolour
[a, xj, with y,,
[o, A]o with y0.
which contradictsq(G) - q + I.
Thus, let «3 - [a, xa]3 be the edge of colour yi incident to a. Let ya - g{ea).
"" >a ■ yi. or ya - y0, terminate the sequence. Otherwise, consider an edge
«a of colour ya incident to a, etc.
In general, if an edge ek - [a, x*]* is found, and if yk ■ #((■„) belongs to
{Voi Vi,.... yjd }, terminate the sequence. If
y* »* Vo« Vi n-i •
wc have yk e Ca because, otherwise, y* Ca, yk ■ g(ek)$ CKk, and we can
recolour
[a, x4]* with y*.
I«.**-i]*-i with y*_,,
■ ■tii
[a, b]0 with y0,
which contradicts q(G) - q + I.

258
GRAPHS
Since graph G is finite, this process will terminate with an edge ek
[a, xk]k such that
Consider the sequence
[a, b]0, [a, x,],, [a, x2]2 [a, xk]k
of distinct edges. Vertices b, x,,xa,.... .v* are not necessarily all distinct;
however, it follows from (ii) that xk x,, since ek e, and
g([a, xk]k) - g{[a. Xj]j).
3. Let fi be a colour of C6 - Ca (since 9 > f/t;(fl), this colour fi exists by the
Uncolourcd Edge Lemma). Consider the connected component of graph
G(P, yt) that contain vertex a. This component is an alternating chain or
colours y, and fi. and one of its endpoints is vertex a (because/? $ Ca). Denote
this chain by ney, [a. z]. At most one of xlt x, can be on this chain because
Yi 4 Cx, and y, CXk\ besides, if x, (or xk) lies on this chain, it is an endpoint.
Flu. 12.3
Case I. y, y0 and z xt.
Thus, x, f neri[a, z].
So, colours >'/ and (I can be interchanged along chain /<«,, [a, z] to form *
new colouring with y, i Ca and y, i CX(. We can then recolour
[a, Xj]j with yj,
[a.Xj.^j-i with Yj-i,
[a.x,], with ylt
[a, b]0 with y0.
This contradicts q(G) - q + I.

C HKOMAT1C 1NDPX 259
Case 2. y, »* yoandz - x,.
Thus xk 4 fig,, [a. z).
Colours y, and (I can be interchanged ulong this chain to form a new colour-
colouring with y, i CB and y, 4 CXk. We can then rccolour
[a,x*)k with y;,
(a, **-,]»_, with yk-i,
[a,xk.2]k.2 with y»_2.
[o, Xj+iX
[a.xj]j
[a.Xj.A
[0.*ih
[a,b]0
in w>t^
f+i with
with
,_i with
with
with
Yj+i.
P.
Vi.
Vi.
Yo-
Tins contradicts q(G) - q + I.
Case 3. y, - Vo o«</ z b.
Thus, A ^ fie,0[a, z].
Colours y0 and /? can be interchanged along chain ne,0[a, z]. We can then
colour [a, b]0 with y0. This contradicts <y(C) - q + I.
Case 4. y, - y0 an</ r - A.
Colours y0 and (I can be interchanged along chain nB,0[a, b]. We can then
rccolour
[a, xk]k with >'O,
[a,xt_,]»_, with yt_lt
[a, x,], with y,
[a, b]0 with /?.
This contradicts q(G) - <7 + I.
Since we find a contradiction in each case, the theorem follows.
Q.E.D.
Corollary 1. Let G - (X, E) be a mulligraph without loops, and let
do(x) + max ma(x, y).
"*
'hen
q(G) < max

260 GRAPHS
Let G' be a partial graph of G with
q(C) < q - max d&x)
and with the maximum possible number of edges.
If G' G, then there exists an edge [a. b]0 in graph G - G'. Let
G" - G + [a, b]0 .
For each x e X, we have
da(x) + ma(a, x) < dfa) < <7,
da~(x) < </c(x) < ^(x) < <y.
Hence, from Theorem 6. the edges of G" can be coloured with q colours. Th«
contradicts the maximality of G'.
Q.E.D.
Corollary 2 (Vizing's theorem [ 1964]). Let G he a multigraph without loops
multiplicity max*.,, mv(x. y) — p. and with maximum degree h. Then
Clearly,
da(x) - da(x) + max mc(x, y) < h + p
Thus, from Corollury 1.
q(C) < max d*(x) <h + p.
Q.ED.
Corollary 3. Let G he a simple graph with maximum degree h; then q(C)
equals h or ft + \.
The proof follows by setting p - I in Corollary 2.
Corollary 4. Let G he a multigraph without loops and with maximum degree
h — 3; then q(G) equals 3 or 4.
If G contains two vertices x and y such that mo(*. y) - 3, then a connected
component of G contains only these two vertices, and its edges can be coloured
with only three colours. Therefore, we may assume that wo(x, y) < 2 for all
x, ye X. By removing an edge between each pair x, y with ma(x,y) ■ 2,
simple graph G' with maximum degree < 3 is formed, and. from Corollary
q(G') < 4. Since each edge of G - G is adjucent to at most three colours,
can be coloured with the missing colour.
Q.E.D-

CHROMATIC INDEX 261
theorem 7. (Ore [ 1968]). l/G is a multigraph without hops and with maximum
h. then
q(G) < max { h, max | = (do(xj) + do{x2) + </c(xj)) 1.
I lx,.*i.x,] L * J I
^tre the inner maximization is taken over a/I elementary chains [*,, xa, xa] of
Clearly, the theorem is valid for graphs with I, 2, or 3 edges. If the theorem
is true Tor all graphs with m - 1 edges; we shall show that it is also true for a
graph G with m edges.
Let
q - max | h, max [^ (</0(x,) + dc(xl) + do(x3))] J
If edge [a, b]0 is removed from G. then the remaining multigraph G' can be
coloured with q colours because
q > max j h', max [i (do.(x,) + </0.(x2) + «/0-(*j))] } •
We shall assume that q(G) - q + I and we shall show that this leads to a
contradiction.
I. Since q > h, the Uncolourcd Edge Lemma yields
| Ca - Cb | - q - da(b) + I > h - do(b) + 1 > 0
I C» - C. | - q - do(a) + 1 > h - do(a) + 1 > 0.
Thus, let a e Co - C», /? e C» - C«. Let at be vertex joined to a by an edge
!<•■ Oilo with colour a.
2- We shall show that C«, => Cb - C«.
The connected components of the partial graph G(a.P) generated by
colours a and ft consist of either isolated vertices or alternating even cycles
coloured a and /?, or alternating open chains coloured a and p. Since a 6
C« - Cb, peCb- Ca, the connected component of G(a,P) that contains
VCftex a is neither an isolated vertex nor an alternating cycle. Therefore, it is
"i open chain with endpoint a that contains ax. Furthermore, b is the last
^Hex of this alternating chain because, otherwise, by interchanging colours
9 and p along this chain, colour a could be removed from set Ca and [a, b]0
c°uld be recoloured a. Thus P e C. for each p e Cb - C«. Hence
C, =>Cb-C..

262 ORAPH3
3. We shall show that Cai => Ca - Cb. Suppose that this is not so. Let •
Ca - Cb and a' CM. Let a2 be the vertex joined to a by an edge with coloUr
a'. We see, as in Part 2, that the connected component of the partial
Flu. 12.4
G(a'. P) that contains a is an alternating chain with endpoints a and b Thus
the connected component of G(a\ (i) that contains a, is an alternating open
chain /<[a,, z] that contains neither a nor b. Clearly, z er,. because ft e C,.
Colours a' and /? can be interchanged along chain yi[er,, z]. Then [a, a,]0 can
be recoloured with fi, and [a, b]0 can be recolourcd with a, which contradicts
q(G) - q + 1. Hence
C, = Ct - C..
4. From Parts 2 and 3,
C, => (Ca - Cb) u (Q - Ca).
Mence
da(oi) - I Cai | > | C. - C, | + | Cb - C. | >
>(q-dc(b)+ 1) + (q - dc{a) + 1).
Thus
which contradicts the definition ofq.
Q.E.D.
Corollary (Shannon [1949]). Let G be a mulligraph without hops and w"*
maximum degree h. Then
m < ft) ■
The proof follows immediately from Theorem 7.

CHROMATIC INDEX 263
Remark 1. The above corollary can also be deduced from Vizing's theorem.
■ «t h > 0, and suppose there exists a multigraph G with maximum degree
r3 hi
/, such that q(G) - I — I + I . By removing some edges if necessary, we
^gy assume that G is critical with respect to this property, i.e. if we remove
from G any edge, the resulting graph has chromatic index [-=-
C contains two distinct vertices a and b with
because, otherwise, mo(x, y) < J - I for each pair x, y, and the Vizing theorem
vwjuld imply
«<»«*♦ [£]-[¥]•
Let [a, b]0 be an edge joining a and b. The multigraph C - G — [a, bH
can be coloured with q - [— 1 colours (because G is critical); for one such
colouring,
\C. u C\ < (dc(a) - mc{a, b)) + {dc{b) - ma(a, b)) + (mc{a, b) - 1)
< dG(a) + dc(b) - mo(a, b) - 1
Thus edge [a, b]0 can be coloured with one of the q colours, which contra-
contradicts i(G) - q + I.
Remark 2. It is easy to show that the bound f -?- 1 given by the corollary
can be attained for each value of A. Consider the multigraph GH consisting of
three vertices a.b,c, = edges between a and b and = edges between a and
f and between b and c.

264
GRAPHS
1*M
FIR. 12.5
(Sec Fig. 12.5). Each edge of G» requires a different colour, and the total
number of edges is
»■[¥!
»♦«-[¥]
if h -2k
if
h-2fc+ I . ,
Thus G,, is a multigraph with maximum degree h and chromatic index
[¥!•
Vizing [1965] has also shown that if a multigraph G with maximum degree
h docs not contain a subgraph GH as defined above, then
3/t
«»«[¥]•
If h is even, this result can be improved by the following theorem:
rhcorcm 8. Let G be a multigraph wit/tout loops and with even maximum
degree h — 2 k. Let 2 < p < k. If G do s not contain Gap as a subgraph, then
3/» r h
*«<¥-[&]•
Two cases must be considered:
Case 1: p - k.
Let G be a multigraph with maximum degree < 2 /c such that q(G) — 3 ■
We may assume that G is critical with respect to this property. It remains U>
show thut C is identical to the graph Ga» shown in Fig. 12.5. G contains two
distinct vertices a and b with
mo(a,
k.

CHROMATIC INDEX 263
, otherwise, the Vizing theorem would imply
q(G) < max dc(x) + max mc(x, >»)<2fc + (fc— l) — 3fc— 1.
tfhich contrndictsq{G) - 3 & .
Let [a, b]0 be nn edge joining these two vertices. Let
C - G - [a, bH .
Multigrnph C can be coloured with 3 Ac — 1 colours, and for such n colour-
colouring.
}k- 1 -|C,,uC»|<
< (dQ(a) - mc(a, b)) + (dc(b) - ma(a, b)) + (mc(a, b) - I) -
- dc(a) + dc{b) - mc(a, b) - 1 <
<2* + Ik - k - I - 3k - I .
Thus
mc(a, b) - k .
Similarly, grnph G' contains a vertex bx ndjncent to b with
m(r(h, />,) > &
because, otherwise,
max (dc.(fc,) + mc.(fc, b,)) <2k + (k-l)-3fc-l.
which by Theorem 6 implies that q(G) < 3 k - I, which contrndicts <?(G) -
3*.
Since mo(h, a) - mc(a, b) - I - k - I, then ft, o. Furthermore, since
rf.(&) < 2 * and w(X*. a) - * , we have
"'c(*i ft|) - * •
By applying the snme argument to the pnir b, bt, we see thnt G contains a
vwicx ha bt with mcF,, ba) - k, etc. Vertices a, b, ft,, h form n chnin
°ffc-tuple edges, thut cun terminate only at vertex a. Therefore, G hns n cycle
P such thnt each pnir of two consecutive vertices is joined by k edges. Since G
■s criticnl, G is connected and. therefore, G can have no other edges.
If cycle n hns even length, then <j(G") ■■ 2 k (from Theorem 5), which con-
contradicts q(G) -3 k.

266 ORAFHS
If cycle n has odd length n0 > 5, then, from Theorem 5,
which contradicts q(G) — 3k.
Thus, cycle \i hns length 3, and G - Gak .
Case 2:p < k.
Let G be a multigraph with mnximum degree 2 /c that docs not contain a
subgraph Ga p. We shall show that G enn be coloured with 3k 1-1 colours.
By adding vertices, G can be imbedded in a regular multigraph of degree
2 k that contnins no subgraph Gar; thus, we may nssuinc that G is regular.
From the Pctcrscn theorem (Theorem 2, Ch. II). the edge set £ of G can
be decomposed into k factors £,, £a £k. The grnphs //, — (X, £,) are
regular of degree 2: thus«(//,) < 3, and, consequently.
- Y q(H,) < 3 k .
Note that, if /i is even, this is Shannon's result.
Let F- £, u £a u—u £p. Multigraph (X, F) has maximum degree 2/>
and contains no subgraph Gap. From Cnsc I, (A\ F) has chromatic index
< Ip - I. Thus
Q.E.D.
Corollary (FiamCik. JucoviC [1970]). Lei G be a multigraph without loops
and with maximum degree h. l/G does not contain a partial subgraph G, (set
Fig. 12.5). then
If h - 2 k is even. then, from Theorem 8 with p - 2,

CHROMATIC INDEX 267
If h m 2 A: + 1 is odd, then G can be embedded in a graph G with maxi-
maximum degree h' — 2 k + 2 that contains no subgraph G«. Thus
it»
Q.E.D.
The table below gives upper bounds for the chromatic index of a multi-
•rapli G of maximum degree h. These bounds are calculated from the corol-
corollary to Theorem 7 and from Theorem 8.
h - 3 q(C
/i - 4 q(C
/i - 5 <?(C
/i - 6 9((
/> - 7 <?(C
A - 8 q(C
r)< 4
?)< 6;
!)< 7;
?)< 9;
?)<10;
?)< 12;
if
if
if
if
if
and if
G ^ G* t q(G\ ^ 5
C t G3, ?(G) < 6
G 4> C?6, f(C) < 8
G 4> G7, 9@ < 9
G ^ Gg, q(G) < 11
G + G4, ^(G) < 10.
We conjecture:
Conjecture. Let Gbea mulligraph without loops and with maximum degree h.
l/G $ G,, wAere 4 < j < A, r/>en
If both h and j are even, the conjecture becomes Theorem 8. If/1 - s is odd,
the conjecture can be proved as in Case 1 of Theorem 8.
Another conjecture is the following:
Conjecture (Vizing [1965]). Let Gbea simple graph with maximum degree h.
Then the vertices and edges ofG can be coloured with h + 2 colours so that no
too adjacent vertices have the same colour, no two adjacent edges have the
same colour, and no vertex has the same colour as an edge incident to it.
3- Edge colourings of planar graphs (abstract)
From Corollury 3 to Theorem 6, we know that if G is regular of degree 3,
then q(G) < 4. It has been conjectured that the chromatic index of a regular
Planar graph of degree 3 without islhmi is exactly 3.
This conjecture has never been proved. However, by studying colourings of
graphs, many results related to the famous "four colour conjecture"
been discovered:
Theorem 9. h'or a regular planar mulligraph G of degree 3 without isthmi,
'he following conditions are equivalent:
A) the faces of G can he coloured with 4 colours so that no two adjacent
faces have the same c ilour.

268 GRAPHS
B) the edges of G can he coloured with 3 colours so that no two adjact
edges have the same colour,
C) each vertex can be assigned a coefficient p{x), where p(x) equals + i
- 1. such that each face /< satisfies
£ p(x) ■ 0 (mod 3).
A) =» B) If the faces of G are coloured with 4 colours a, /?, y, 6, label with a
" each edge separating an a-facc from a /7-facc or separating a
y-facc from a <5-facc. Label with a " 1" each edge separating an
a-face from a y-facc or separating a /J-face from a <5-facc. Finally,
label with a " each edge separating an a-face from a j-face or
separating a /J-facc from a y-facc.
Two adjacent edges cannot be labelled with the same symbol (since
this would imply that two adjacent faces have the same colour).
Hence condition B) is satisfied.
B) » A) Suppose that the edges of G can be coloured with three colours
0, 1, 2. The edges with colours 0 and I form a regular graph of
degree 2, and, therefore, its faces can be coloured with only two
colours p and q. The edges with colours 0 and 2 also form a regular
graph whose faces can be coloured with two colours r and s.
Thus, each face of G can be coloured with one of the four combina-
combinations pr, ps, qr, qs. If two fuces are separated by an edge of colour 0,
they are coloured with distinct combinations. The same is true for
two faces separated by an edge of colour I or of colour 2. Thus, the
faces can be coloured with 4 colours so that no two adjacent facet
have the same colour.
B) * C) Consider a 3-colouring of the edges with colours 0, 1,2. If the
three edges incident to vertex x are 0, 1,2 in clockwise order, let
p{x) - + I; otherwise, let p(x) - - 1. Now follow the contour
of a face y« in the clockwise direction. Each time a vertex x with
p(x) — +1 is crossed, the index of the colour decreases by one, and
each time a vertex x with p(x) - - 1 is crossed, the index increase*
by one (modulo 3). Therefore, the algebraic sum of all the coeffi'
cients of the vertices on contour n is necessarily a multiple of 3.
C) =» B) Suppose that the coefficients p(x) satisfy condition C).
Starting from some arbitrary edge to which we assign the label 0,
we shall now give each edge u label, either 0 or I or 2: this is done

CHROMATIC INDBX 269
step by step with the help of the coefficients p(x) in such a way
that the three edges incident to a vertex x are 0, I, 2 counter-
counterclockwise if p(x) - + I. and in the reverse order if p(x) - - I.
If G is connected, each edge [a, b] will be given a label g(a, b).
It remains to show that this labelling is consistent. Let v be an ele-
elementary cycle which contains a set F of faces. Let xe X{ if vertex x
is incident to exactly / faces in F. Let e(x, v) - - I for x e JC,,
+ 1 for x e X2,0 for x e X3. If [a, b] and [b, c] arc two consecutive
edges of v in a counter-clockwise tour, then
g{b, c) m g(a, b) + e(x, v)p(x) (mod. 3).
Moreover, we have (modulo 3),
2 «(*. v)p(x) - | (/ 2 Pix)) - T I />(*) ■ 0.
in 1-0 \ irtXi / jirP *•»
The consistency of the labelling on v follows.
Q.E.D.
Corollary 1. If the number of edges bounding each face is a multiple of 3,
then the edges of the graph can be coloured with only three colours.
Let p(x) — + I for each vertex x: condition C) of the theorem is satisfied.
Q.E.D.
Corollary 2. If the number of edges bounding each face is a multiple of 2
ihcn the edges of the graph can he coloured with only three colours.
Let v be an elementary cycle, that contains faces nit na,... .Thelength /(v)
of v satisfies
" (mod. 2).
This implies that v has even length; therefore each cycle has even length,
from Theorem D, Ch. 7), each vertex x can be assigned a coefficient
■ + 1 or - I so that no two adjacent vertices have the same coefficient.
Nearly, these coefficients satisfy condition C) of Theorem 8.
Q.E.D.
Conjecture. For a simple planar graph with maximum degree h > 6, the
chromatic index equals h.
For h — 2, 3, 4. S, it is easy to construct a simple planar graph G with q{G)
* h + |. For h > 8, the conjecture was proved by Vizing [1965).

270 GRAPHS
EXERCISES
1. Consider the complete graph X,, where n is even. Denote its vertices by 0,1,..., „ _ .
and consider the function
f(a,b)-a + b mod (ft—1) if a,b + n—\,
•»2« mod(n—1) if «n*n—l,*-n—I,
b mod (ft—1) if a - n— l,b »* n— I.
Show that /defines a (n - D-colouring of the edges, i.e.,
A) 0 «/(«,*)« n -2.
B) /(a. ft) is an integer for a * b,
C) /(«, *) - /(fc,«).
/fc
2. In a bridge tournament. Ap players simultaneously play on p tables, and no pUyer
wants to have the same partner twice. Show that it is possible to organize Ap - I game,
so that, for a permutation/on the set of chain, each player seated in chair x will p|(y
his next game in chair /(*).
3. Let C be a simple connected graph that decomposes into two connected component!
Ci and Cj when vertex a Is removed. If d is the subgraph of G generated by Ci v-» (o|,
and If Ci Is the subgraph of G generated byCjUft), show that
q(G) - max { q(G\). tf(Gi), da(a)} .
4. Let mi > m2 > — > m, be a non-increasing ?-tuplc of integers. Consider the relation
(m|) •< (m,) defined by
£ »»i < J) m< (* ™ 1, 2 ^ — I)
If m, > m, + 2, the f-luple (mj) defined by
ma ■■ m* (At ^ /»/)
is called the transftr o/(m.) by (IJ). Show that a transfer of m defines a f-tuple m ■<
m, and that each f-tuplc m* with m" •< m can be obtained from m by a finite number of
transfers.
5. Let mi > mt >•••> m, be a non-increasing sequence of integers, and let G be •
multigraph with
edges coloured ai, a, a,, where (he number of edges coloured a, equals m,, for alU
Using Exercise 4. show that it On',) is a sequence with (m',) < (m,), then G can be colourco
in q colours with m\ edges coloured a,, for all I.
Hint: Note that in the partial graph </(<>,. o,) generated by the edges with colour «
and a;, a connected component is an alternating open chain whose extremities ht»
ditTcrent colours. The interchange of colours «, and a, along this chain defines a transfer
(Folkman. Fulkerson

CHROMATIC INDEX 271
t show that a simple regular graph of degree h with order has chromatic index equal
*+ I-
• ut G be a simple graph with maximum degree A such that q(G) - h + I, and such
■hit <?(c - *) ■ * for every edge e. Show that:
I. Each vertex adjacent lo a vertex of degree k Is adjacent to at least A - k + I
vertices of degree A.
2 C possesses an elementary cycle of length > A + I. .
(Vizing 11965])
I Let C be a connected simple regular graph of degree 3 with m edges and with q{G) - 3.
Colour the edges a. j9. y or 0 ("no colour") so that the three edges incident to the same
vtriex are cither all uncoloured or each has a different colour or one is uncoloured and
((other two have the same colour.
If edge t, is coloured t,, then the colouring defines a vector t - fa, ta »B). Any
c louring that satisfies the above conditions is called a good colouring.
Define an operation + (called "sum") as follows:
O + O-a+a-/( + /)-y + y-O.
O + a-a+O-«, O + j»-/f + O-j», O + y-y + O-y.
fi + Y-V+fi-*, y + «-« + y-/>, « + /J-/J + «-y.
Show that:
A) a good colouring * is characterized by the properly
£ «-0 (xeX)
B) the vector sum of two good colourings is also a good colouring,
C) the set of all good colourings together with the operation +, is an abclinn group,
D) if C is planar and without isthmi, a set of generators of this abelian group consists
of the union of a fundamental basis of cycles coloured with a, and of a fundamental
basis of cycles coloured with p.
(Vigneron [1946], [1939], [1961])

CHAPTER 13
Stability Number
1. Maximum stable sets
Consider a simple graph C - (X, £). A set S c X is defined to be stable
if no two distinct vertices of S are adjacent. In other words, S is stable if,
and only if,
rc(S) nS-0.
Let y denote the family of all stable sets of C. Then,
0ey
Sey.AcS m. Aey.
The stability number «(G) of G is defined to be the maximum cardinality
of a stable set,
<x(G) m max \S\.
Example I (Gauss). Problem of the eight queens. Can eight queens be
placed on a chessboard so that no queen can capture another queen? This
famous problem is equivalent to finding a maximum stable set of a simple
graph G with 64 vertices and with y e I'a(x) if squares x and y arc in the same
row or in the same column or in the same diagonal.
8
6 6
5 *' 5
4 I 4
3 J V 3
2 2
I * I
G2631485) FIS28374)
Fig. 13.1
272

STABILITY NUMBER
273
his problem is more difficult that it would appear at first sight, and Gauss
bc'icvcd that it had only 76 solutions; in 1854, Schachzeitung, a
jf chess journal, gave only 40 solutions. In Tact, there arc exactly 92
lutions, which can be obtained from the following permutations:
G2631485)
C5841726)
A6837425)
E1468273)
F1528374)
D6152837)
E7263184)
D2751863)
E8417263)
E7263148)
D8157263)
C5281746)
Each or these permutations gives a possible diagram (sec Fig. 13.1) and
from each of the diagrams we can obtain eight solutions by rotating the
chessboard and by reflecting the chessboard with respect to the principal
diagonal. The last permutation C5281746) gives only four distinct solutions
because it yields the same diagram after rotation of the chessboard.
Recently, it has been shown that it is always possible to place k queens
on h k x k chessboard for all k > 4 (HofTman. Loessi, Moore [1969].)
Example 2. Covering a chessboard with tetraminos. Consider the prob-
problem of covering the 64 squares of a chessboard with 16 tetraminos, of various
shapes, each covering exactly 4 squares. For example, if there arc IS L-shaped
Ktrnminos and I squure tetramino (Fig. 13.2), we can see that no covering
is possible by the following argument: if the squares of the chessboard are
coloured black and white as in Fig. 13.2, then the number #i| of white squares
and the number n3 of black squares covered by an L-shapcd tetramino always
satisfies /i, - na ■ 2 modulo 4.
fa.
ft
'ft
fa.
ft
ft
fa
V<
ft
ft
v/a
7/,
Y/<
V/a
ft
v/a
ft
v//<
ft
Va
ft
ft
ft
V/.
ft
ft
{.•shaped
tetramino
square
letramino
Fl8. 13.2

274 ORAPHJ
In general, the problem reduces to verifying that the stability number or
certain graph G equals 16. To each tetramino / together with the position)
occupied by this tetramino, there corresponds a vertex x(i, X). Join vertjo*
*(/, A) and x(k. v) either if / - k, or if / * k and the set of squares occupied
by tctrumino i placed in position k overlaps with the squares occupied by
tetramino k placed in position v. Each stable set with 16 vertices of the
resulting graph corresponds to a covering of the chessboard with 16 tctrj.
minos, and vice versa.
Numerous analogous problems have been proposed by Golomb [1965],
The construction of a maximum stable set is a particular case of the prob.
Icm of linding a minimum transversal set of a hypcrgraph, which is discussed
in Chapter 18.
Consider a simple graph G - (X, E). A clique is defined to be a set C X
such that each pair of distinct vertices in C arc udjuccnt. Let - (Clf C,,
.... Ck) be a partition of X into cliques. Let
0(G) - min |»|
v
denote the smallest possible number of cliques that partition X.
Theorem I. Let G be a simple graph then s(C) < fl(G). Furthermore JfSlsa
stable set anil if is a partition into cliques such thai | S | - | |. then Sis a
maximum stable set anil V is a minimum partition.
For each stable set S and each partition ' - (C\, Ca Ck), we have
|SnC,|<l (/-I, 2 k).
Hence. |5| < |' |. Therefore
a{G) - max | S | < min | <€ \ - 0(G).
Furthermore, if | So \ - | 01. the above inequality implies that
|S0|-a(C). |«Jfo|-0(C).
Q.E.D.
There are many classes of graph G with a{G) - 0(G). (Sec for instance the
graphs considered in Examples I and 2.) Other classes of graphs with this
property are studied in Chapter 16.
Remark. Given a minimum partition - (Ct, C8 Cfc) for a graph G
with <x(G) - 0(G), it is easy to construct a maximum stable set by a back-
tracing procedure: Select any vertex a-| 6 Ct. Choose a vertex xaeCa tri»
is non-adjacent to .t|. Choose a vertex xaeCa that is non-adjacent to ■'
jfa, etc. When this procedure can no longer continue, backtrack (as in th*

STADIIITY NUMI1I K 275
vp|oration of an arborcscencc described in Chapter 3). Proceed until a
ini stublc set has been found.
Given a simple graph G - (X, £) and u stable set S <= X, an alternating
relative to S is defined to be a sequence
O -
of distinct vertices alternately belonging to A - X — S and to B - S that
satisfies the following conditions:
A) fl,e/4.
B) b,eB- {bltha 6,_i)and
fc(fy)n {fli.fli a,} * 0.
C) a,^sA - {fli.«j a,} and
ro(al+l)n{bl,bi b,} »* 0,
ro(fl/+i)n{fl,.flj fl(} - 0.
An alternating sequence is said to be maximal if no more vertices can be
added to it without violating B) or C). We shall sec that for the maximum
stable sets, alternating sequences play a r6lc similar to that of alternating
chains for the maximum matchings.
Recall from Theorem D, Ch. 7) that a graph without odd cycles possesses
a partition (A. B) of its vertices into two stable sets A and B\ such a partition
is called a bicohmring of the vertices.
I emma I. Let G be a tree, ami let {A. B)bea bicohuring of its vertices with
I B | > | A |. Then, graph G has at least one pendant ivrtex in B.
Suppose that all pendant vertices of G arc in A. Let Ax c A be the set of
pendant vertices. I ct fl, c B be the set of pendant vertices of the tree
Gjr-,,,. Let Aa c A be the set of pendant vertices of the tree Gx.Al.Bl,
etc.
We have | <4, | > | Bx |, because each pendant vertex of GX.M is not
Pendant in G, and can be mapped into one of its neighbours in Ax by an
injection. Similarly, | fl, | > | A2 |, etc.... Thus, we can write:
I *x I > I Bx | > | A21 > | B21 > - > | B, | > | /J,+ I |,
0, »* 0; «,+1 - 0 ■
i 0- wc have
*h'ch contradicts that | A | < | B \.

276 ORAPHS
If/),., ■ 0, then set Bq reduces to a single vertex (that is not pendant j
). Thus. \A, | > | Bv |. and
M I - I I At | > I | B, | - | B |.
r-i /-i
In both cases, a contradiction results.
Q.E.D.
Lemma 2. L t G he a tree ofonhr n. and let {A, B) be a bicohurlng ffy
vertices such that M|-|/?|or|^|-|fl| + l. Then, there is an alternat.
ing sequence (ax, bt, aa, ba ) that uses each vertex ofC exactly once.
1. Clearly, the lemma is true for n - I and n ■ 2.
2. If the lemmu is valid for n - 2 k, we shall show that it is also valid Tor
n - 2 k + I. Since |<4|-|#| + l>|fi|, there exists by Lemma I a
pendant vertex ak*i in A. For 6'A.,OvM), there is an alternating sequence
(fl,,/;, />„) that uses ull vertices, by the induction hypothesis. Hence
(fl,, />, bk, aytl) is the required alternating sequence.
3. A similar argument shows that if the lemma is valid for n ■ 2 k + I,
then it is valid for« ■ 2 A: + 2.
Q.E.D.
Theorem 2. A stable set B is maximum if, and only if, there Is no maximal
alternating sequence of odd length.
1. If such un alternating sequence a exists, then B is not a maximum stable
set, since B' - (B — a) U (a — B) is a stable set with greater cardinality.
2. Let A be a maximum stable set, and let B be a stable set with | B \ .
| A \. We shall show that there exists a maximal alternating sequence of odd
length relative to B. Let Bo - B - A, and let Ao - A • B. Hence. | Bo
| Ao \. Let AiV Bi, Aa\J B2 Akv Bkbc the connected components of
subgraph G,,ouBo, where A, c a0, B, c fl0 for / - I, 2 fc. Thus,
f |fl,|.|fio|<Mol- Z M.I.
r.
Hence, there exists an index /with | Bt \ < \ A, \. Let / - I be this index.
If | fl, | + I - i Ai |, a spanning tree of the connected subgraph GMvii
has a bicolouring {Au /?,). By Lemma 2, its vertices constitute »
maximal odd alternating sequence a relative to Bx. Clearly, sequence a ■*
also for C an alternating sequence relative to B\ furthermore, sequence e
is a maximal alternating sequence because if A B — a, cither be B - A
Bo and b is not adjacent to any a,eo, or be Bn A and b is not adjacent to
any ateo (since A is a stable set).

STAHIIITY NUMBFR 277
If | Si | + I < | Ai\, then pendant vertices can be removed from the
ipanning tree or GAl^Bl until the remaining tree has a bicolouring (A\, #,)
with | Bi | + I - \A[ |. by Lemma I. Then, as before, its vertices constitute
ide required sequence a.
Q.E.D.
Theorem 3. A vertex of C is defined to be "free" if it belongs to at least
maximum stable set but not to all maximum stable sets. Let B be a stable
Ki. A vertex x is free if, and only if, there is an alternating sequence o ■
(fll, bx. a o,, A,) relative to B that contains x and such that
beB-a *- rc(b) n { a,, a2 a,} - 0.
1. If such a sequence a exists and if xeo, let
A - X - B,
B' - (B - a) u (a - B),
A' -(A - a) u {a - A).
B' is also a stable set, and since | B' \ - | B |, B' is a maximum stable set.
If xe B, then x $ B'; if * £ B, then * e B'. Thus, vertex x is free.
2. Let B be a maximum stable set that contains .v, and let A be a maxi-
maximum stable set that docs not contain *. Let Bo — B — A. Let Ao — A — B,
Clearly, xe Ro.
In the subgraph generated by Ao V Bo, let /l,u Bx denote the connected
component with
Ax c Ao , Bt c Bo, xeBx.
If | fi, | < | Ai |, it is easy to sec that there exists in G a maximal alter-
alternating sequence of odd length relative to B (as in the preceding theorem).
This contradicts that B is a maximum stable set. Similarly, | fl, | > | Ax \
contradicts that A is a maximum stable set. Thus
l*i I-1/1,1.
Consider a spanning tree of the connected graph generated by /),uA,.
From Lemma 2, its vertices constitute an alternating sequence a — (ait bit
°a bq) of even length in graph GAnuBtl. Sequence a is also an alternating
"cjuence of even length in graph C.
Furthermore, if be B - a, cither be B - A - Bo, which implies that
Veicx b is non-adjacent to {altat aq), or beBnA, and b is non-
'djacent to { a,, a a,} since A is a stable set. Thus, sequence a is the
requircd alternating sequence.
Q.E.D.

278 ORAPHS
2. The Turin theorem and related results
No good algorithm is known for determining the stability number of.
graph; however, there arc several known bounds for a(C) in terms f
other invariants.
Theorem 4 (J. C. Meyer [1972]). Let C - (X, E) be a simple graph of
n with vertices *i, *a xn such that
If for an integer p. 2 < p < n,
d(AxK) + •• + to-na) <n - p,
then each stable set with fewer than p vertices is contained in a stable set withp
vertices.
The theorem will be proved by an induction on p.
If p - 2, and if d^xm) < n - 2, then da(x) < n - I for all *£ X, and
each vertex belongs to a stuble set with two vertices. Thus the result is true.
Let p > 2. We shall assume that the result is true for p, and we shall show that
it is also true for p + I.
Suppose that
MX*) + - + dtAXn-f.J < ft - I - I .
Then, a fortiori,
doC*..) + - + ddxm-f >3) *n- p- \ < n- p.
Hence, from the induction hypothesis, a stable set 5 with fewer than p vertices
is contained in a stable set So with p vertices. Let So - {yx yT). Then,
ddyi) + - + diAyp) < d,Axn) + - + d<Axn.,n) < n- p- I .
Therefore.
I Wo) \Kn-p-\X-So\.
Consequently, there exists in X - So at least one vertex that is not adjacent
to 50. Therefore. S is also contained in some stable set with p + I vertices.
Q.E.D.
Corollary (Bcrge [I960]). In a simple graph C with order n and maximum
degree h, each maximal stable set has cardinality] T at least.
_ l/i + IJ
,^) < h(p - I).
The definition of p implies that h(p — I) < n — p.

STABILITY NUMBER 279
,us
MxH) + d^Xn-x) + - + </„(*,.,**) *n-p
t d the corollary follows from Theorem 4.
Q.E.D.
Example. There are 99 bridge players. No player will play with someone
u'tiom he does not know. Let A be the maximum number of players unknown
,, the same player. For what values of A can a bridge table (four players) be
farmed?
Let the vertices of C represent the players, and let the edges of C join
pairs of players who do not know one another. If It - 32, then
*
-I33J -3
ind no table of four is possible. On the contrary, if h - 31, then a table can
be formed. In fact, given any three players who know one another, a fourth
can be found to play with them.
Remark I. Clearly, the above corollary implies that
I or each n and for each h < n, we shall show that this inequality is the
tai possible.
I. We shall show first that if a and b are two integers, then
We can write:
a - qb + r, 0 < r < b, q - [^J .
lr'1 • 0. then a - I - (q - \)b + (b - I), 0 < b - I < h. Therefore,
i- > 0, r < b, then a - I - qb + (r - I). 0 < r - I < b. Therefore.
2. Let
■LttI ■If+tj
Since n - I - (p - IXA + I) + r, 0 < r < h + I, a graph with n vet-
can be formed from a clique with r + I > 0 vertices and /» - I disjoint

280 ORAPHS
(h + l)-cliques. Clearly this graph has maximum degree h since, for all e
(even for /> - I), it contains a (h + l)-clique. Furthermore, its stablitv
number equals p.
Remark 2. Related to Theorem 4 is a conjecture, due to P. ErdOs [19571
that was recently proved by Hajnal and Szemeridi [1970]: '
The vertices of a graph of order n and maximum degree h can be partitions
into h + I disjoint stable sets each with \ ;■ "■ ■ 1 or [ t-^-t 1 vertices.
Remark 3. For planar graphs, the lower bound on a(C) given by Remark I
seems weak. The following conjecture suggests an improvement:
Conjecture. (P. Erdtts). Every planar graph C of order n has stability number
This conjecture is weaker than the famous "four coloilr conjecture",
which states that the vertices of a planar graph can be partitioned into 4
stable sets S1( 53, S3, S«, since this implies:
It can be shown that ct(G) > ? for each planar graph C of order n. See the
corollary to Theorem A2, Ch. 14).
Theorem 5 (Turdn [1941]). Ifn and k are two integers, n > k > 0, put
let r be an integer such that:
n -k(q- I) + r, 0< r*k.
Let C.i, be the simple graph that consists ofk disjoint cliques, of which r hat*
q vertices and k — r hare c — I vertices. Then, every graph C with n verticts
and stability number < k that has the minimum possible number of edges It
Isomorphic to GK%k,
I. Consider the various vulues for n given below:
fc + 1 2fc + l 3/c + l ... qk+ I
k + 2 2k + 2 3fc + 2 ... qk + 2
2 k 3fc 4fc ... (q + I) k ...

STABILITY NUMBER 281
The proof is immediate for the values of n in the first column; we shall
j)iow by induction that if the theorem is true for the 9-th column, then it is
true for the (q + l)-st column, i.e. for graphs with order
n - k(q + I) + r, 0 < r < k.
2. Let C - (X, E) be a graph of order n with a(G) < k, and with a
minimum number of edges. Hence a(G) - k. Let S - {Si,sa,..., sh} be a
stable set with k vertices. Each vertex xe X — S is adjacent to 5, since,
otherwise, «(C) > k.
Subgraph Gx.a has order n — k and stability number < k\ hence, by
the induction hypothesis,
m(Gx.s) > m(G,,.kmk).
Since GKik can be formed from C,_ft.k by adding a vertex to each of the
disjoint cliques in Gm.kik,
«(CB|4) - m(GH.kik) - n - k.
Furthermore, since /»(G) < '»((/„.*) by the definition of G, it follows that
n-km\X-S\* mG(X - 5, 5) -
- m(G) - m{Gx.s) < ^(C,,^) - m(C,,.»i4) - n - k.
Hence,
m(C) - m(Gm*),
_s) - m(Gm.kik),
and, by the induction hypothesis,
Gx-s ■ ^«-m •
Thus, C is formed from k disjoint cliques Ct, C3 G and one stable
sets.
3. The above inequalities show also that
I*-S| -mo(X-S,S).
Thus, each vertex xe X — S is joined to 5 by exactly one edge. Let s(x)
denote the neighbour of x in S.
If x e C, y 6 C,, i j, then s(x) ? s[y) since, otherwise, the set
{x,y}u(S-{s(x)})
*ould be a stable set with k + I vertices. lfxeC,,yeCt, then s(x) - j(,y)
$'nce, otherwise, the number of cliques C, would be > | S | - k. Thus, graph
c 's isomorphic to graph Gm „.
Q.E.D.

282 ORAPHJ
Corollary 1. If G is a simple graph with n vertices and m edges, and iw
a(G) - k, then '*
where q ■ [ -r 1 . Equality holds if, and only (f, G is isomorphk to graph G
Graph Gn.k consists of r cliques of q vertices and (k — r) cliques of q . |
vertices. Thus the number of edges equals
„,«;„.)■
- ~— (r9 + kq - rq - 2 k + 2 r) -
(q- \)(n-k + r)
2
-i((/- \){n - k + h - fcfa- I))-
Q.E.D.
CorollHry 2. If G is a simple graph with n vertices and m edges, then
Equality holds if, and only if, the connected components of G are cliques with
the same cardinality.
I. If «(C) - k, n - k(q - I) + r, then from Corollary I,
m > ^ fa - 1) (n - k + r) - yr (n - r) (n - k + r) .
It is easy lo sec that (« - x)(n - k + x) has its minimum value in [1.1ft
only for x ■ k. Hence
2 km > (n - k) n.
or
*- "'
2 m + n '
2. If C consists of p cliques with cardinality n0, then a(C) - /j, and
_-! £i^ P.a(C).
2m + n pno(no - I) + pn0

STABILITY NUMBER 283
fi equality holds in the inequality of the corollary.
Conversely, if equality holds, then G - Gn.k, and r - k. Thus, G has the
rtnuired form.
Q.E.D.
Corollary 3. If G is a simple graph with n vertices and m edges, then
Equality holds if, and only if, each connected component of G is either a 2-
clique or a 3-clique.
We may assume that G is connected (otherwise, the result could be demon-
demonstrated for each connected component).
1. If G is a 2-clique, then
1 - a(C) > ^y- - I .
and equality holds throughout.
If C is a 3-clique. then
I - ct(G) > ^-^ - 1 ,
and equality holds throughout.
2. Let G be a simple connected graph with n vertices and m edges. First
suppose that m > n. From Corollary 2,
2 n\ + n '
Note that the inequality
n1 2ii- m
2 m + n 3
'« equivalent to
n2 - 3 mn + 2 m1 > 0
or
(n - m) (n - 2 /«) > 0 .
if wi > n, we have
Equality can hold only if in - « (because wi ■ s is not excluded since
^ ") and if G is a clique. Thus, equality holds only if C is a 3-clique.

284 ORAPHS
3. Now, suppose that I < m < n; then m - n - 1 because G is con.
nected. Thus, G is a tree. Since G is bipartite and n > 2,
n n + 1 2n - (n - 1) In - m
>~ —
Equality holds only if
i.e. if n - 2, i.e. if C is a 2-clique.
4. Finally, suppose that m - 0 and n - I; then, clearly,
. _. In — m
ct(G) > 5 .
QE.D.
Corollary 4 (Zarankicwicz [1947]). Let G be a simple graph with n vertices
and with maximum degree h, and let k - [ " ]■ Then, a(G) > k. Besides,
If G does not consist of k disjoint cliques each with cardinality^-, we have
• ft.
The number m of edges in G satisfies
2 m - £ do(x) < hn < ("■ - l) n.
Hence, from Corollary 2,
If G does not consist of k disjoint cliques with the same cardinality, then,
from Corollary 2.
Q.E.D.
Remark. We shall show that Corollary 4 implies:
If n is a multiple of h + I, let ft -. " .. Then.
n + i

STABILITY NUMBIR 285
If n is not a multiple of h + I, let k - [ " ■ 1. Then,
L n + 1J
In both cases,
Q.E.D.
3. a-Critlcal graphs
Graph G is said to be a-critical if each partial graph C obtained from G
by the removal of an edge satisfies a(G') - a(G) + I. See Zykov [1949].
An a-critical graph is necessarily a simple graph.
Property 1. A graph G with a(G) - k has an a-critical partial graph H
with «(//) - k.
If a(G) - k, any graph H obtained from G by the removal of an edge
satisfies a(H) - k or k + I. Successively eliminate the edges of G whose
removal does not change the stability number until no more such edges
exist. The remaining graph is a-critical and is a partial graph of G.
Q.E.D.
Property 2. In an a-critical graph G with «(G) - k,for each vertex x, there
is a stable set Sx with \ Sx | - k - 1 whose union with xisa maximum stable
set. Set Sx is called a "cell" ofx.
1. Let x be a vertex of G. If there exists a vertex a adjacent to x, then the
removal of edge [a, x) creates a stable set S with k + 1 vertices. Thus, a,
x 6 S. and the set S - {a. x} is a cell of x.
2. If x has no neighbours, each n aximum stable set 5 contains x, and
therefore, S - {x} is a cell of vcrtc*. x.
Q.E.D.
Property 3. In an a-critical graph, two vertices a and b have a common cell
If. and only if, they are adjacent.
Let G be an a-critical graph with a(C) - k.
1. If a and b arc adjacent, the removal of edge [a, b) creates a stable set 5
*ith k + I vertices. Thus. S - { a, b) is a cell of both a and b,
2. If a and b arc non-adjacent, they cannot have a common cell 5 because
then S\j{a,b} would be a stable set with k + I vertices.
~~1 Q.E.D.
Property 4. Let G be an a-critical graph with a(G) - k. If So is a stable set
whh k — I vertices, then the set Ce of vertices that have So as a cell is a clique;
furthermore, Co is disjoint from So and no edge goes from Co to So.

286 ORAPHS
The proof follows from Property 3.
Property 5. In a connected, a-critical graph C of order > 3, each
has a degree > 2.
Suppose that there exists a vertex a with da(a) < 2. Vertex a cannot be
isolated since G is connected. Therefore, a is pendant and there is a vertex
x with
[a,x)eE, </«,(«)-I.
Since n > 3 and G is connected, edge [a, x] is not an isolated edge, and
therefore x has a neighbour b 1> a.
Let Sbx be a cell of both b and x. Then, | Sftx| ■ k — I, and a$Sbx
(since a is a neighbour of x). Vertex a is not adjacent to Sbx, since d^a) m |,
But, then, Sbx u {a, b} is a stable set with cardinality k + I, which contra,
diets a(G) - it.
Q.E.D.
♦- Property 6. Cleen two adjacent edges [a, b) and [b, x] in an a-critical graph,
there is a maximum stable set So such that
A) a, b4S0,xeS0,
B) only edge [b, x] joins b to So.
Let Sbx be a cell of both b and x (which exists b'y Property 3). and let
So - Sfc, u { x }.
Since a $ So (because a is a neighbour of b and is distinct of x), and since
b$ So, Property (I) follows. B) is immediate.
Q.E.D.
The structure of a-critical graphs has been extensively studied. First, it
has been proved by Beineke, Marary and Plummer [1967] that in an a-critical
graph, any two adjacent edges lie on a common odd cycle; Andrdsfai [1967)
has proved that In an a-critical graph each non-isolated edge lies on an odd
cycle without chords. The following theorem generalizes both of these re-
results:
6 (Berge [I970]). In an a-critical graph G with «(G) - *, any
two adjacent edges [a, b), [b, x) He on a common odd elementary cycle without
chords.
t
I. If edge [b, x] is removed from G, a stable set Sbx with cardinality
k + I is formed. Let B - Shx y {/>}. Clearly. B is a muximum stable set
in G, and a, bf B, x B. Only edga [b, x) joins b to B.
if,

STAItlLITY NUMBER 287
2. Clearly, stable set B is not maximum in the partiul graph G - [a. b\\
therefore, by Theorem 2, there exists in C — [a, b] a maximal alternating
jequencc
a ■ (fl|,blt aa,b3,.,.,a,),
with a, 6 X - B and fc, e B for ull /. Since set T - {B - a) u (o - B) is
a maximum stable set in C — [a, b], we have a,beT. Hence, a.bea- B.
The subgraph of C - [a, h) generated by a is connected and has bicolour-
ing (a r\ B,a — B). Let p be the shortest chuin connecting a and b. Since
a,bea — B, n together with edge [a, b] form an odd elementury cycle with-
without chords in G. This cycle contains [a, b] and [b, x] (since only [b, x] joins
b to a n fl).
Q.E.D.
^Corollary 1. <4 connected u-crltlcal graph has no articulation vertices.
Suppose that a is an articulation vertex. Let B and C denote the connected
components obtained by removing a, and let [a, b] and [a. c] be two edges
with be B and ceC.
No elementury cycle can pass through these two edges. This contradicts
Theorem 6.
Q.E.D.
Corollary 2. No clique of a connected ^-critical graph is an articulation
set.
Suppose that Ao is a clique that is an articulation set. We shall show that
the graph is not a-critical. Let A c Ao be a minimal articulation set. Then,
I A | > 1 since, otherwise, the graph is not a-critical by Corollary I. Let B
and C be the two connected components resulting from the removal of A.
Each vertex a e A is adjacent to both B and C (otherwise. A — { a} would
^e an articulation set, which contradicts the minimality of A).
Let [a, b] and [a, c] be two edges with be B and ceC Each elementary
cycle that contains these two edges contains a vertex a' e A distinct from a.
Since A is a clique, this cycle has a chord [a, a'].
From Theorem 6, this contradicts the hypothesis that the graph is <x-
critical.
Q.E.D.
Corollary 3. A connected a-critical graph G is either a clique, or contains
°n odd cycle of length > 5 without chords.
If G is not a clique, then there exist two non-adjacent vertices a and b.
Let [o, x,, x2,.... b] be a shortest chain connecting a and b. Clearly, xx a, b.

288 ORAPHS
No triangle contains both [a, X|] and [xi, x9]. Thus, from Theorem 6, therei*
an odd cycle of length > 5 without chords.
Q-E.D.
The following theorem characterizes a-critical graphs G with a(G)«
(KG).
Theorem 7 (Bcrgc [I960]). For an ^-critical graph G - (X, E\ the follow,
ing conditions are equivalent:
A) the smallest number 0(G) of cliques that partition X satisfies
0{G) - «(G),
B) graph G consists ofa(G) disjoint non-adjacent cliques;
C) each cycle of length 5 in the complementary graph G - (X, JX)\E)
has at least two chords.
A) => B) Let 0{G) - it, and let d, Ca Ck be k cliques that partition X.
Suppose cliques C\ and C3 are adjacent and let fli e C,, a2eCa
with [ai. o3] e E. 1 he graph H obtained from G by removing edge
[fli, fl3], satisfies
«(//) - k + I .
Since (Clt C2,.... Ck) is also a partition into cliques, it follows that
0(H) < k.
Thus,
k + I - «(//) < 0(//) < k,
which is a contradiction. Thus, condition B) holds.
B) => C) Let /J - [xi, x2, x3, x4, xe, xj be a cycle in C. We have:
[x,, x,], [x2, Xi), [x3, x4], [x4, x,), [x,, x,] e 2(X) - E.
Let C,, Ca,..., Ck be /c disjoint non-adjacent cliques that form
graph C. We shall show that /! has at least two chords in G.
If k - I or 2, cycle /J cannot exist; therefore, we may assume
that k > 3. Cycle ju encounters at least three different C, succes-
successively; for example, suppose
x,eC,, x, eC,, *3eCj.

STABILITY NUMBER
289
We may assume that cither vertex x« or vertex x8 belongs to C2
Otherwise, fl would have two chords in C and C) would hold.
-—T
xA s C2 v, c C,
FIr. 13.3
For example; suppose x«eC2 (see Fig. 13.3). Then [xi,x«] and
[x9, xs] are two chords of fi in C, and C) holds.
C) a- (I) First, we shall show that if a, x, y are three distinct vertices ofC
with
[a,x]eE, [a,y)eE,
then [x, y) e E.
Suppose that [x, y] i E, and consider a common cell Sax of a
and x and a common cell 5av of a and y. (From Property 3, these
cells exist.)
Since [a, y]eE, y$Sax. Similarly, x*S0V. Cell SaK is not a
cell of y, because [x, y]i E and Property 3. Let b be a vertex of
Sax that is adjacent to y. Similarly, let c be a vertex of Sav that is
adjacent to x (see Fig. 13.4).
Edge of G
Edge of C
Ftg. 13.4

290
oraphs
We have b j* c, because one of these two vertices is a nu
of .v and the other is not. Therefore, [x. y, c, a, b. x] is an clctnen.
tary cycle of G of length 5, and has at most one chord in 17, which,
contradicts C).
Thus, the binary rclution defined by "x is adjacent to or identical
to y" is an equivalence relation. The classes of this equivalence
relation are disjoint, non-adjneent cliques of C. Hence, <x(G) m
(KG).
Q.E.D.
Theorem 8 (Hajnal [1965]). In an ^.-critical graph C without isolated ve^
tices, each stable set S satisfies \ r^S) | > | S |.
If | S | - I, the result is evident, since C has no isolated vertices. If the
result is vulid for all stable sets with fewer than q vertices, we shall show that
it is valid for a stable set S with q > 2 vertices. Let
S ■ { X], .Vj,..., Xq }.
Let
Oq ™ { X], Xj, •.., Xf_| / .
Suppose | roE) | < | S |. Then, by the induction hypothesis, we have
q - I - I So | < | ro(S0) | < | ra(S) | < q - I .
Hence, T0E) - roEo). Furthermore, we have
I rc(A) I > M I (A
50) .

STAND ITY SUMMER
291
Thus, from the KSnig-Hnll theorem (Theorem 5, Ch. 7), So can be matched
into rn(-S0). und the q - I vertices of ro4S0), say ax, oa, .... a,, i, satisfy
[x,,a,]eE (i- 1,2 q - I).
Since ro(x,) 0 (because G has no isolated vertices), there exists a ver-
vertex of f(S0) - ro(S) that is adjacent to x,. Without loss of generality, let
fll be this vertex; let D be a common cell of a, and xt. Then, au Xi $ D
and x9<t D (because x, is a neighbour of ax). Thus, set D can intersect set
5U l'a(S) only at the following vertices:
x9 or
x3 or a3; xt at
x,_,or«g.,.
Ttiercfore,
If a(C) - k, the stable set T- D-(Su ToE)) satisfies
| r| - | D | - | E |
and. hence,
Furthermore, set S u T is stable (because T n ra(S) - 0). This contra-
contradicts s(G) - k.
Q.E.D.
Lemma. Consider a graph C with no stable set of cardinality k + I. If
S\,Sa, ..., Spare stable sets with k elements, then
US,
i-i
n ^
>2k.
If/; - I, the result is evident. If the result is valid for/; - I stable sets, we
shall show that it is also valid for /; stable sets Si, 59, .... Sp.
\-'or k - 1.2....,/;, let
Ak - U S,
Clearly, set flP_, is non-adjacent to set /4P-i; thus, no vertex of set

292 orapiu
MP_ i n 5P) \J Bf-t'is adjacent to another vertex of the set and, consequents
the set is stable. Hence
A) \(Ap.lnSp)uBp_l\*<x(G)mk.
Furthermore,
B) lAp.tnSp\misp\-lSp-A,.t\m
- | S,| - \A, - /<„-, | - k - \AP | + | Ap.x |.
By comparing equations (I) and B), we have
| *,_, u (/<„_, n 5P) | - M,_, n 5P | <
Since the induction hypothesis yields that | Ap.i | + | /?„_, | > 2 k, \
I 0,1 - I «,_. n S,| - | *,_, | - | *,_, - 5, | >
>|^-,|-|«,_, -M,-inSp)|>
> B* - | Vi 0 " (Mrl " I Vi I) - 2* - Mf |.
Q.E.D.
Theorem 9 (Hujnal [1965]). Let C be an a-crltlcal graph without Isolated
vertices with «(C) - k and \ X \ - n. Then
dc{x)*n-2k+ I (xeX).
Suppose there is a vertex a with | rG(a) | > n - 2k + I. We shall show
that there exists a stable set So with | roE0) | < | 50 |.
Let Si,Sa, .... 5P be the stable sets with cardinality k that contain vertex
o. If we let A - U 5,, fl ■ f] S,, then, from the lemma,
Furthermore, each vertex xe A is non-adjacent to set B — {a}. Besides,
if ye F(Aa), there exists a common cell Scv of a and >> (from Property 3).
From the definition of B,
, {
Thus, 5ai/ u {,y} is a stable set that contains B - {a}, and again, >> is non-
adjacent to set B - {a}. Finally, by letting 50 - fl - { a},

STABILITY NUMBER 293
inds
rc(S0) <= X- A - rG(a).
Hence,
l/V,E0) | < n - \A | - | ro(a) |<n-B*-|«l)-("-2*+l)-
1 -|*|-I-|SO|.
Thus, set So is stable and satisfies | r^S0) | < | 50 |, which contradicts
Theorem 8.
Q.E.D.
Corollary I. Let C be an a-critical graph without isolated vertices. Then
a(G) < x • Equality holds if, and only if, each connected component of C Is a
2-clique.
Let k - a(C); then each vertex .v satisfies
I < da(x) < n - 2 k + I .
Hence,
n-2k >0.
Equality can hold only if each vertex has degree I, i.e., if all edges of C
are isolated edges.
Q.E.D.
Corollary 2 (Erdos, Gallai [1961]). An a-critical graph C with a(G) - k
and | X | ■ n contains at least 2 k — n isolated vertices.
If C has n isolated vertices, then the result follows because k - <x(C) - n.
If C hasp > n isolated vertices, the graph C obtained from C by removing
the isolated vertices satisfies
ri _ n - p ,
«(C) -k' -k-p.
From Corollary 1, graph C, which is a-critical and without isolated ver-
vertices, satisfies
0 < n' - 2 k' - (n - p) - 2(k - p).
Thus p > 2 k - n.
Q.E.D.
Example I. Consider a graph C consisting of an odd cycle with n - 2 k +
1 vertices. Then «(C) - it, and G is a-critical. This graph is the only connected
'■critical graph with n - 2 k - I (because, from the Hajnal Theorem,
dc(x) < 2 for all x, and neither an open chain nor an even cycle is a-critical).

294 GRAPHS
Example 2. Consider the graph G with four vertices a, b, c, d and 6 chain
J"i[a. b), jua[o, r], juj[o, d], >«,[/;, c), n&[b. d), no[c, d] of odd length > 3 ^
are vertex-disjoint (except at their endpoints). Let s(G) ■ k: then
n - 2 k - 2 .
As seen from Fig. 13.6. the removal of uny edge [x, y] increases the stability
number, and, therefore, G is a-critical. Andrdsfai [1967] lias shown that
A-7
n-16
Fig. 13.6
the graphs of this type are the only connected K-critical graphs with n — 2 k - 2.
The structure of connected a-critical graphs with n — Ik ■ 3 is not known.
So far, only the a-critical graphs with connectivity 2 have been charac-
characterized (Wessel [1970]). Several methods for constructing new classes of
a-critical graphs have been described by Plummer [1967], Gallai and others
(see George [1971]).
The following result gives the maximum possible number of edges in
a-critical graphs.
Theorem 10 (ErdOs, Hajnal, Moon [1964]). IJ G is an a-critical graph with
n vertices and m edges, and with s(G) - k, then
m
;("*♦')■
Equality holds if, and only if, G consists of a stable set with k — I vertices
and a clique with n — k + I vertices.
Let fh{n) denote the maximum possible number of edges in a graph C
with the properties:
A) G has at least one stable set with cardinality k,
B) the removal of any edge creates a new stable set with cardinality k + '•
C) G has order n.

STABILITY NUMiBR 295
|. First, we shall show that
,. s ^(n - k+ 1\
Note that
A(k + 1) - 1
because each graph satisfying A), B) and C) with k + 1 vertices and with a
maximum number of edges consists of a 2-clique and a stable set of k — 1
vertices.
Consider a graph G that satisfies A), B) with order n > k + 1 and with
/*(») edges. Let a and b be two adjacent vertices. From B), the removal
of edge [a, b] creates a stable set S with cardinality k + 1, and aeS, beS.
Consider the graph C obtained from G by removing vertex b and each
edge [a, z] with z 4 r^b). For C, set S - { b ] is stable with cardinality k.
The removal of an edge of C creates a stable set with cardinality
k + I, since this creates in C a stable set with cardinality k + I that either
contains a and not h or contains neither a nor b. Hence the number of edges
in C is
') </*(«- I).
Furthermore, the number of edges removed from C to form C is
m(G) - m(G') <l + |jr-5|-l+ji-(ft+l)-H-
Hence
/»(«) - m{G) < m(C) + n-k </,(« - I) + (n - k).
Thus,
+ 0- l
Hence,
-fc+l) (ti-k + l
AW < 1 + 2 + ... + („ - k) .OzJ_ |
2. Denote by Ck(n) a graph consisting of the union of a (n - k + I)-
li and a stable set of k - I vertices. Clearly, graph C(n) satisfies (I),
B)
and C), and has (" " k + ')edges.

296 oraphs
Thus
n-k+\
and Gk{n) is a graph with the maximum possible number of edges.
3. We shall show that each graph of order n with fk(n) edges that satisfy
(I) and B) is isomorphic to G*(n). Clearly, this is true for n - k + |, jy
induction, we shall show that it is also true for a graph G of order n > k +■ |f
Consider an edge [a, b] of C, and construct a graph C by removing vertex
b and each edge [a, z] with z £ ro(/>). As before, m(G) - m(G') < n . k.
Hence
m(C) >fk(n) ~{n-k) -fk(n - I).
From Part I, C satisfies conditions (I) and B) since G has order n - I and
has the maximum possible number of edges, it follows from the induction
hypothesis that
G'm Gk(n- \)m /rB_4u5»_,,
where Kn.k is a clique with n — k vertices and Sk.x is a stable set with
cardinality k — I. We shall consider two cases.
Case \: a Sk.i (sec Fig. 13.7).
Since the removal of edge [a, b] creates in G a stable set with cardinality
k + I, there exists a vertex y with
ye Kn.k. y 4 ro{b) , ro(y) n Sk., - 0.
Furthermore, if ce K*.k and c y, then c is not adjacent to a because,
otherwise, a stable set with cardinality k + I could not be created in G by
removing edge [a, c]. Furthermore, c is adjacent to b, since the number of
edges removed from G to form C is equal to
A(n)-fk(n- I) -n-k
Fig. 13.7

STABILITY NUMBER 297
Then, if edge [h, c] is removed from G, a stable set with cardinality k + I
ggpnot be created, which contradicts the hypothesis. Thus, Case I cannot
occur.
Case 2: aeKn.k.
Vertex b cannot be adjacent to set Sk.x in G (because the removal of
t. b] would create a stable set with cardinality k + I). On the other hand, b is
jdjaccnt to each vertex of Kn_k (because the number of edges removed from
qio form C equals n — k). Thus, G is the union of an (n - k + l)-clique,
Kn.i,u C*). and a stable set Sk~x.
Q.E.D.
4. Critical vertices and critical edges
This section generalizes some of the results of Section 3. A vertex x of G
\p,b] should create a stable set with cardinality k + I). Also, b is adjacent
to each vertex of #„_* (because the number of edges removed from C to
form C equals n — k). Thus G is the union of an (n - k + l)-cliquc
A'n-«rv{b) and a stable set S*_,.
1 hcorcm II. 7>w endpolnls of a critical edge are non-critical vertices.
Let G' be the graph obtained from C by removing a critical edge [x,>].
Then,
ot(G') - ot(G) + I .
Hence,
«(C) > a(G*_(J(J) - a(G'x.{xt) > <x(C) - I - <x(G).
Therefore,
«(GX.{X]) - <x(G).
and vertex x is not critical.
Q.E.D.
Theorem 12. 77w c</;rc incident to a critical vertex are non-critical edges.
Let x be a critical vertex of G, and let [x, >'] be an edge incident to x.
Then,
Let Ex denote the set of edges incident to x. Then,
a(G) < a(G - [x. y\) < a(G - £x) - 1 + a(G^_tJI,) - <x(G).
a(G - [x, y]) - a(G), and edge [x, y] is not critical.
Q.E.D.

298 ORAPHl
Theorem 11 shows that the only critical vertices in an ot-critical graph a,
the isolated vertices. The following theorem applies this result.
Theorem 13. If a graph G with order n has no critical vertices, then «(G) , ?
Furthermore, if equality holds, then
<x(G) - 0(G) - ».
Suppose a(G) > ^; we shall show that <x(G) - 0(<7) - ^.
Since G has no critical vertices, each vertex of G belongs to the comple.
ment of a maximum stable set. After removing an edge without chang.
ing the stability number, this remains true a fortiori.
Suppose that enough edges are removed to form an a-critical graph H
with a(H) — a(G). Then, graph H has no critical vertices (and no isolated
vertices) and satisfies ot(//) > -.
From Corollary I to Theorem 9, «{H) - -, and // consists of ^ pairwise
disjoint edges. Hence,
Therefore
<x(G) < 0(G) < 0(H) - ~ .
Q.E.D.
Corollary. If, for each vertex x, the subgraph Gx - i*j contains a stable sti
S with \S\ ■> -, then G contains a stable set So with
I So | > \ + I .
From Theorem 13. G hns at least one critical vertex a. Hence
>i
2
-,.,) + I > J + l •
Q.E.D.
5. Stability number and vcrtcx-covcrlnjjs by paths
In this section, we shall assume that G is a I-graph
Theorem 14 (Gallai. Milgram (I960]). In a \-graph G - (X, U), there exl
a(G) elemtnfary paths that partition X.

STABILITY NUMBER 299
Let M - { Hi,n nk} be a family of elementary pairwise disjoint paths
ttiat cover X. Let A(M) - {olt a , ak} be the set of initial vertices of the
paths in M. A family A/ always exists since all paths of length 0 can be
chosen.
We shall show that there is a family M' of elementary paths that partition
X such that | A/' | < «(G) and A(M) c A(M). (This proposition is in fact
stronger than the theorem.)
1. Clearly, this proposition is true for a graph with one vertex. If it is
valid for all graphs with less than n vertices, we shall show that it is also valid
for a graph G with n vertices.
2. If | M | > ot(G') + I, we shall show that M can be replaced by a family
17 with | A7 | < <x(G) + I and A(K!) c A(M). Consider the subgraph gener-
generated by Xi ■ X — Hi. The stability number of this subgraph is
and A/j ■ {ft ju*} partitions the vertex set of GXi. Hence, from the
induction hypothesis, there exists a family A?, in Gx, such that
< «(G). /KM,) c
Thus, family Kf - M\ vj { ^,} is the required family of G.
3. Clearly, if | fif \ < <x(G). the proposition holds.
If |A7|-a(G)+l, set /!(#")- {a,.a3....} is not stable, because
l^(A7) | > «(G). Thus, there is an arc joining two of its vertices; let this
arc be (a,, aa).
If the path jui e M which begins at ax has zero length, then the proposi-
proposition holds. Otherwise, let
£i - [aXtbx,Cu ...]•
The subgraph generated by XY - X — {a) possesses a family A/, with
|.W,| < «(C7), ^(A/,) c {/>,,fl,,fl,...}.
(^■p » <"i •<h\
FIR. 13.8

300 ORAPIH
If fc, e A{Mi) or if oa e A{Mi), the proposition follows by adding verte* 0
to one of the paths in Mx. l
If bt i A{Mi), a2 $ A(Mi), then | A/, | < «(C), and the proposition follow,
with
M' - A/, u [a,].
Q.E.D.
Remark. This result is the best possible since for each simple graph
it is possible, by assigning a direction to its edges, to form a I-graph G with
mini A/| - «(<7).
M
To find this direction, simply choose a maximum stable set So and direct
each edge incident to So towards So. Then, | M | > »(G) for all families A/,
Corollary 1 (R6dci). A complete I-graph contains a hamiltonian path.
Since «(G) - I for a complete graph G, the existence of a path that covers
all the vertices of G follows from Theorem 14.
Q.E.D.
Corollary 2 (" Di I worth's theorem"). I/G is a transitive \-graph andifM
a family of paths that partitions its vertex set, then
min | M | - «(C).
Theorem 14 implies that min | M \ < <x(G). Since G is the graph of a
partial order, each path generates a clique. Therefore, a(G') < (KG)
min | M |, and equality holds.
Q.E.D.
Application ("Sperner's theorem"). // X is a set of n elements, and if
is a family of distinct subsets of X such that no member of' is contained
in another member of- , then
This famous result has recently been extended in several directions (Erdtis
and Katona, Klcitman, Mcshalkin. etc). We shull show that the Spencer
theorem is an easy consequence of Corollary 2.
Consider the graph G* whose vertices represent the different subsets of X
Place the vertices of G into n + I rows 0. I. 2,.... n where the h-\\\ row con-
contains the vertices corresponding to the subsets with n — h elements. There
are L\ such subsets. Join vertex a to vertex b by an arc (a, b) if the corres*
ponding sets A and B satisfy A = B.

STABILITY NUMBER 301
The maximum cardinality of a family & with the required property
equals the stability number a(G). Since the set of elements in a row is stable,
(here exists a stable set with I r/M I members in row h - I -1
prom the Dilworth theorem, this stable set is maximum if and only if
I n \
the vertex set of G can be partitioned into I , „, I paths.
Mill
To show that this is possible, notice that, given two consecutive rows, the
smaller can always be matched into the larger: Since two consecutive rows
form a bipartite graph in which all vertices of the same class have the same
degree, this follows from Corollary 4 to Theorem E, Ch. 7). For all h, let Eh
be a maximum matching between the rows h and h + I. Then the union of the
£„ define the edges of the required paths.
Q.E.D.
EXERCISES
1. Using the Turan theorem, show thai the minimum number of edges in a graph C
with n veriices and stability number k equals
(Las Vergnas)
2. If n > A(p + 1), and if each subgraph CA with \A | - 2/> + I satisfies \A \ -
«(CA) < p, then show that m « [ -]• (Erdos, GalUi A961))
3- Consider a simple graph C - (A". £) and a partition (*,. X2 *,) of X such that
I-1
Show that o(C) - 0(G) if
- tKGx.) (/-I. 2 p).
4> If the partition (A"., Xt A",) above has more than one class, then C is said to be
decomposable. Show ihat a connected a-critical graph is not decomposable.
(Plummer)
'• For a non-decomposable graph C that is not a clique, show ihat a(C) < 0(G).
'• Let C ■ (A", £) be a connected graph, and let Eo be a maximum matching. Suppose
'hat Eo is not a perfect matching. Let Xi denote the sel of veriices thai can be reached

302 ORAPHS
from an unsaiurated veriex by an even alternaiing chain (and by no odd alicmM|
chain). Lei X% denote ihe sei of vertices thai can be reached from an unsaturaied vtru*
b dd li hi (d b li hi) Sh h if
ed v
by an odd alternating chain (and by no even alternating chain). Show that if JKi u jf
X, ihen A", is a maximum siable set of graph G. (Berge (I937T
7. If A c X is a set of vertices in G, let S(A) denoie a maximum siable sei of the 11,1.
graph generaied by A. Given a maximum matching £0, let Mi, M3 denote thecoT
nccied components of ihe subgraph generaied by the "mixed" vertices. Let A,/.
denote the connected components of the subgraph generated by the "inacccsiible«
vertices. Let
If there is at most one connecied component of mixed vcriices, show thai 5 jt t
maximum stable set of G, (Berge [19S?])
8. Show that in a graph G without critical vertices, | ra(S) | > 151 for every stable set £
9. Recall that if G is an n-critlcal graph without isolated vertices then G has no critical
vertices. Show that the converse is not true by considering the following graphs:
A) Tt, the complementary graph of a cycle of length 7 without chords, s
B) C« augmented with a vertex that is joined to each vertex of C».
10. If G is a graph with n vertices and m edges with m > I " 1, show that G contain* a
triangle.
Show that there exists a graph of order n with | " ledges that contains no triangle*.
11. Let A - {a0. t>\ aP} be an articulation set of graph G. If. for an Integer * with
0 < k < p. each vertex a0, o\ «»■* adjacent to vertices x\ and x3, belonging to dif-
different components of Cx-4, and if each vertex ak,l% a*.2 a, is adjacent to a0, then C
is not o-critical.
(Wessel 11970])
12. Show thai ihe complementary graph of the Pctersen graph (Fig. 10.11) is a-criikal.
13. Let G be a l-graph. Show ihat iherc exists a stable set S such ihat each vertex of C
is me iniiial endpoini of a path of Icngih < 2 wilh a terminal endpoint in S.
(L6vasz. Chvaial)

CHAPTER 14
Kernels and Grundy Functions
I. Absorption number
For a l-graph G - (X. f), a set A <=■ X is defined to be absorbant if for
each x $ A,
f\x)r\A ¥• 0.
Let j/ denote the family of all absorbant sets of graph G. Then
, A' = A -•• A'es/.
The absorption number of graph G is defined to be
fi(G) - min | A | .
This section studies the absorbant sets with minimal cardinality.
Example I. Radar stations. A number of strategic locations x,,Xj, ...
(called cells) arc kept under the surveillance of radar. Radur in cell v4 can
survey x, orxaorx3, as shown in l-'ig. 14.1. Similarly, cell.va can be surveyed
by radar located in x3 or xB, etc.
What is the minimum number of radar stations needed to survey all the
cells? It is the absorption number of the graph in Fig. 14.1, which is equal to
2. since radur stations placed at x3 and *« are sufficient.

304
GRAPHS
Example 2. Chessboard controlled by five queens. What is the
number or queens that can be placed on a standard chessboard so that etch
square is controlled by al Icust one queen ? Five queens arc sufficient because
the absorption number (I(G) of the graph G, defined by the moves or a queen
on a chessboard, is 5 (sec Fig. 14.2).
Note that the same placement or the 5 queens also controls a 9 x 9 chess,
board. The plucemcni at the right in Fig. 14.2 shows how five queens can cot),
trol an 11 x II chessboard. No general results are available for the tnaxi-
mum size n(k) of a chessboard that can be controlled by k queens.
H 35
J_
■
Fig. 14.2
Finding a minimum absorbant set is a special case of the minimum trans-
transversal problem, for which constructive algorithms are given in Chapter 18.
Mere, we shall give bounds on the absorption number.
Proposition 1. If G is a \-graph with n vertices and m arcs, then
p(G) > n - m
Lei A be a minimum absorbant set. Each vertex of X — A is the initial end-
point of an arc going into A. Thus,
„ _ | a | - | X - A | < m.
Hence,
/?(G) - \A | > n - m.
Q.E.D.
Proposition 2. If G is a \-graph without loops with n vertices, then
P(C) < n - max rfj(x).

KIRNILS AND OKUNDY FUNCTIONS 305
Let x0 b< a vertex with <l£(.v0) - max,,, rfj(.v). Since the set
A - X - rj(.v0)
i5 absorbant, we have
(!(G) < | A | - n - max rfj(.v).
x
Q.E.D.
If G is a simple graph, its dominance number /?*(G) is defined to be the
absorption number /?(C) of the symmetric graph G* obtained by replacing
fjch edge in C by two oppositely directed arcs. As in the Tunin theorem, we
shall calculate the maximum number of edges possible in a simple graph with
a given order and dominance number.
1 hcorcm 1 (Vizing [ 1965]). IfG is a simple graph with n vertices, m edges and
P*(G) m k > 2, thin
For each n and each k, equality holds if and only IfG Is ixomorphic to a graph
Gn.* obtained by removing from a (n — k)-ilique the edges of a minimum
covering and by joining its vertices to each vertex in a stable set of cardinality k.
I. First, we shall show that
m < ^ (" ~ *) (« - * + 2).
Clearly, this inequality is satisfied Tor n - 2. We shall assume thai it is
satisfied for all graphs with order < //. and we shall show that il is satisfied for
a graph G with order // > 2 and wilh /?*(G) - k > I. Let G* be Ihc sym-
symmetric graph obtained by replacing each edge of G by two oppositely directed
arcs. Let x0 be a vertex with maximum degree. From Proposition 2. we have
| rc{x0) | - rfj.(xo) - max dj.(.v) < n - k .
Thus, we may write:
| ^c(*o) | - it - * - r ; 0<r<«-Ar.
Ut 5 . X - { x0 } - r^x0) then
|S| -* + #■- I.
lr>'6 /"c(x0). the set (S - ToO1)) u { xo,y) is absorbant. and, therefore.
| 5 - raty | + 2 > k .

306 OKAPHS
Thus,
k + r-\ - \r<:(y)nS\ + 2 > k
or
| I'G(y) n S | < r + I .
Furthermore. ifGs is the subgraph generated by set Sand if A is a minimum
ubsorbanl set of graph C,, then
\Au{xo}\>k
(because A u { v0} is absorbant in G), Hence,
(i'(Gs) > k - I .
Hy the induction hypothesis, the number of edges in Gs is
»'(CS) < ^ {k + r - 1 - (k - 1)) (k + r - I - (k - 1) + 2) -
- j Kr + 2).
Therefore, the number m((i) of edges in G satisfies
2 m(G) - 2 w(C.¥) + | rc(x0) |
< r(r + 2) + (n - k - r) + (n - k - r) (/• + |) + (ii - * - rI -
- (n - k) (« - fc + 2) - /•(»» - k - r) <
< (n - k) (n - k + 2).
The required inequality follows.
2. We shall show that there exists a graph GH,k with order n and with
0*(GBj „) - k, such that
'"(C^)- [j(n - fc)(n - *+ 2)] .
For k - 2, consider the graph CB.a defined by the union of a set Ai m
{ Xi, .va} and a set Bn.a - { x3 xn) where each vertex of A2 is joine"
with each vertex of Bn .a, each vertex of Bn .a is joined with every other verK*
of Bn. a. except for some pairs of vertices that constitute a minimum covering
offlB_a(sccCh. 7. §2).

KEKNEU AND ORUNDV FUNCTIONS 307
This graph satisfies /?*(G,, a) - 2. Furthermore,
- \(n - 2)(n - 3) - [i(« - 2)]* + 2(n - 2)
Thus. Tor this graph GB.a the required equality holds.
If k > 2, define graph Cn>k by adding to graph Gn.kt2.2 a set of Ac - 2
uolatcd vertices. Then.
>»(CH.k) - m(GB.k«3.a) - [i(» - ft + 2 - 2)(n - * + 2)]
Again, equality holds. The inequalities in Part I can be used to show the
niqueness or graph Cn k.
Q.E.D.
Corollary. If C is a simple graph with n vertices and m edges, then
p'(G) < n + 1 - VI + 2 m .
From the inequality of Theorem I, we have
(« - AJ + 2(;i - ft) - 2 in > 0.
Since h - k > 0. then
« - fc > - 1 + V1 + 2m
or
Q.E.D.
2- Kernels
For a 1-graph G - (X, f), a set 5 c A'is defined to be a kernel if it is both
Hablc and absorbent, i.e. if
ll) xeS * r(x)r,S-0 (stable)
<2) S+0 (absorbant).

308 ORAPHS
Not every graph has a kernel (sec graph in Fig. 14.3), and if a graph
sesses a kernel, this kernel is not necessarily unique (see graph in Fig,
In this section we shall present existence and uniqueness theorems.
FIr. 14.3. Graph without kernels Fig. 14.4. Graph with two kernels
ExAMPLt 1 (Von Neumann. Morgenstern [1944)). The concept of a kernel
was first presented (under the name solution) in gumc theory.
Suppose that n players, denoted by (I). B) (»). can discuss together to
select a point x from a set A'(the "situations"). If player (i) prefers situation
a to situation />. we shall write « >'6. The individual preferences might
not be compatible, and. consequently, it is necessary to introduce the concept
of effectire preference. The situation a is said to be effeciirely preferred to b, or
a > h% if there is a set of players who prefer a to b and who are all together
capable of enforcing their preference for a. However, effective preference is
not transitive, i.e., a > b and b >• c do not necessarily imply that a >• r.
Consider the I-graph (X, f). where f(.r) denotes the set of situation!
effectively preferred to v, Let S be a kernel of the graph (if one exists). Von
Neumann and Morgenstern suggested that the selection be confined to the
elements of S. Since S is stable, no situation of S is effectively preferred to
another situation of S. Since .9 is absorbant. for every situation x $ S. there it
a situation in .9 thut is effectively preferred to x, so that x can be immediately
discarded.
Example 2. Basis of axioms. Consider a "theory", i.e., a set of proposi-
propositions a, b, c thut we shall represent by vertices; add arc (a, b) if proposi-
proposition b implies proposition a. The resulting graph C - (A\ U) is transitive. i.e-:
(a,b)eU, (b,c)eU -»• (a,c)eU.
A basis of axioms of the theory is a set B of propositions (called "axioms")
such that:
A) each proposition not in B follows from one of the axioms,
B) no axiom follows from another axiom.
The problem of finding a basis of axioms reduces to finding a kernel of C. H
will be shown later that each transitive graph has a kernel.

KERNELS AND ORUNDV FUNCTIONS 309
proposition 1- A necessary and sufficient condition for a set S c X to be a
fa I -graph G - (X, f) Is that its characteristic function p,(.v) satisfies
4p,{x) - 1 - max <p,(y).
Recall that the characteristic function <p,(x) of set S is defined by
■ I if x 6 S,
■ 0 if x
m 0i we define
max <p,(y) - 0 .
|. Let S be a kernel. Since S is stable, we have
I ■* xe S ■* max ^,(y) - 0.
yUx)
Since i' is absorbant, we have
<p,(x) - 0 m- x*S *> max <pt(y) - 1 .
Combining these, the required formula follows,
2. Let <p,(x) be the characteristic function of a set S which satisfies the
formula; then we have
< S -» (p,(x) - 1 *■ max (fi,(y) - 0 — r(x) c\ S - 0
i S -»> <p,(x) - 0 -► max <p,(y) - I **■ l\x) c\ S + 0 .
yH»)
fius, S is a kernel.
Q.E.D.
Proposition l.lfS is a kernel, then S is a maximal stable set and a minimal
tisorbant set.
Let S be the kernel of a l-graph G - {X. f). If a f S. the set S <J {a } can-
Jt be stable because f(«) nS 0. If /; 6 S. the set T - S - {6 } cannot
• absorbant because /; * 7" and f(/;) n 7" - 0.
Q.E.D.
Theorem 2. If G — {X, F) is a symmetric \-graph. then G has a kernel,
furthermore, a set S c A' is a kernel if. and only if. S is a maximal .stable set.
• Clearly, a maximal stable set SofG is absorbant (otherwise, there would
**ist a vertex x i S non-adjacent to S, and 5 could not be a maximal stable
*0. Thus. S is a kernel.

310 orapiu
2. Conversely, if S is a kernel of a symmetric graph C, then Sis a maxjm
stable set (because, otherwise, S would not be absorbant).
QE.D.
Algorithms to construct all kernels of a graph have been presented by ROv
[1970] and Rudeanu [1966]. y
Theorem 3. IfG ■ (X. O is a transitive I -graph, each minimal absorbam
set has cardinality P(G). Furthermore, a set S c X is a kerne! if, and only if c
is a minimal absorbant set
Let G - (X, /) be a transitive I-graph, i.e.:
m. zer{x).
Consider the strongly connected components of G. If C is a component
with m<j (C, X — C) - 0 (i.e., no arcs leave C), C will be called a terminal
component. Since the graph obtained from G by contracting each strongly
connected component contains no circuits, G possesses terminal component!
Let C|, C, C, be the terminal components of G. Each of these is a
complete symmetric subgraph, because G is transitive.
1. If A is a minimal absorbant set, then A contains at least one vertex from
each terminal component. Otherwise, A n Q - 0, and ench x e C, satisfies
xiA, r(x)nA m 0,
which contradicts that A is absorbant.
Let a, 6 A n C, for each ;. Consider the set
A' - {ai,a2 a,}.
A' is also an absorbant set because each x A' is the initial endpoint of an
arc leading into A' (because of transitivity). Hence A' ■ A, and each minimal
absorbant set is also a minimum absorbant set.
2. If 5 is a kernel, then 5 is a minimal absorbant set by Proposition 2. Con-
verscly, if A ■ {a,, a2 aq} is a minimal absorbant set, then A is stable be-
because no arc leaves a terminal component; therefore, A is a kernel.
Q.E.D.
Corollary 1. A transitive \-graph has a kernel, and all of its kernels have tht
same cardinality.
•*
Consequently, in Example 2, all the axiom bases have the same cardinality'

KERNELS AND ORUNDY FUNCTIONS 311
forollary 2. In a graph G - {X, U), there is a set B c X such that
A) no paths join two distinct vertices of B,
B) each vertex xi B is the initial endpoint of a path leading into B.
The corollary follows by applying Corollary 1 to the transitive I-graph
IX. f) obtained by creating an arc (.*. y) if there is a path from x to y in G.
Q.E.D.
Theorem 4. A \-graph without circuits possesses a kernel, and this kernel is
unique.
Given a 1-graph G - {X, D without circuits, consider the sets
X{Q)-{x/x*X, l\x)-0}
X(\)-{xfxiX@), r(x)<=X@)}
XB) - {xfx4X@) u X(\). n.\) c X@) u X(\)}. etc...
These sets are pairwise disjoint, and x X(k) if, and only if, the longest
* th from x has length k. Since G contains no circuits, the sets X(k) form a
partition of X.
A characteristic function <ps of a kernel Scan be successively defined on the
«ts Af(O), X{\), X{2), etc. by the equality of Proposition I. Furthermore,
\ (.v) is uniquely defined for each x.
Q.E.D.
The following theorem due to Richardson shows that a graph without odd
circuits possesses a kernel. The original proof of this result was long and in-
yolved. The less complicated proof presented here is due to Victor Neumann.
A semi-kernel of a I-graph G - {X, f) is defined to be a non-empty stable
*t S c X such that each x adjacent to S has at least one successor in S.
Ultima. If for each non-empty subset A c X, the subgraph GA has a semi-
then G has a kernel.
Let S be a maximal semi-kernel of G, and let A - X - S - Fa{S).
If A - 0, then S is a kernel. If A 0, then GA has a semi-kernel T. Sets
5 and T arc non-adjacent: thus. S u T is stable. Each x SvT that is ad-
adjacent to S u 7" has a successor in S u T. Thus, S u T is a semi-kernel of C,
*h'ch contradicts the maximality of S.
Q.E.D.
Theorem 5 (Richardson [1953)). If G - {X. T) ix a 1 -graph without odd
r'rcw//j, then G has at least one kerml.

312 ORAPHS
From the lemma, it suffices to show that G possesses a semi-kernel. We n
assume that G is connected.
Let X{ be a strongly connected component of G with F(Xi) c %x. tf
| Xt | - 1, then Xx is a semi-kernel. If | X{ \ > I. and if xoe Xy, let x^
vertex in Xt distinct from x0. Then, all paths >i[x0, x] remain in the interior
of Xi; and they have the same parity (since, otherwise, an odd circuit could
be formed with a path p[x, x0]).
Let S denote the set of all x e Xx such that all puths ;<[x0, x] are even. Set $
is stable. If z e X is adjacent to S. then each successor of z belongs to S. Thus,
S is a semi-kernel.
Q.E.D.
Note that the graph in Fig. 14.3 has no kernel; therefore it contains an odd
circuit. Similarly, the graph in Fig. 14.4 docs not have a unique kernel,
therefore it contains a circuit.
3. Grundy functions
Consider a l-graph G - (X, /') without loops. A non-negative integer func-
function g{x) is called a Grundy function on G if for every vertex x, g(x) is the smal-
smallest non-negative integer which does not belong to the set {g{y) I ye f(x)}.
This concept originated with P. M. Grundy [1939] for graphs without circuits.
It was extended to l-graphs by Bcrgc and Schiitzcnberger A956).
The Grundy function can also be defined as a function g(x) such that
(I) g(x) - k > 0 implies that for each J < k, there is a ye F(x) with
B) g(x) - k implies that each y f(x) satisfies g(y) k.
A pseudo-Grundy function is defined to be a function that satisfies property
(I) above and
B') g{x) - 0 implies that each y f(x) satisfies g(y) 0.
We shall sec in the following development that a Grundy function (or a
pseudo-Grundy function) determines a kernel of a graph.
Remark I. Some graphs have no Grundy function. Some graphs have
more than one Grundy function. (Sec Fig. 14.5 where the value of a Grundy
function is written next to each vertex.)
FlB. 14.5

KERNELS AND GRUNDY FUNCTIONS 313
Remark 2. If G has a Grundy function g(x), then G has a kernel, since the
set
S-{xfxeX, g{x)-Q}
satisfies simultaneously:
a\ xeS -•• g(x) - 0 m. min g(y) > 0 -* /*(*) n S - 0
a) xiS * r{x)>0 -* mingOO-O -* r{x)nS 0
The converse is not true. It is left to the reader to verify that the graph G in
Fig. 14.6 has a kernel {d} but hus no Grundy function.
'I hcorem 6. IJGisai -graph such that each subgraph has a kernel, then G
possesses a Grundy function.
Let G be such a graph, and let So be a kernel of G. Let 5, be a kernel of
C, - Cx.So: let Sa be a kernel of G3 - Cx-cs^s,). etc.
The sets 5, form a partition of X. Let g{x) be an integer defined by:
g(x) -k >~ x 6 Sk.
We shall show that g{x) is a Grundy function of G.
1. Let g{x) - k; we shall show that for each 7 < k, there exists a vertex
fE/'U) such that g(><) -/
Since x e Sk, and k > J, vertex x is present in graph G,. Since * ^ S,, there
is a vertex y S, such that ye f(.v); thus, there is in F(x) a vertex y with
*(>■)-./.
2. Let j(»(.v) - A; there is no vertex ye F(x) such that g{y) - k. because
'hen Sk would not be stable.
Thus g{x) is a Grundy function.
Q.E.D.
Corollary 1. A symmetric graph possesses a Grundy function.
This follows from Theorem 2.
Corollary 2. A transitive graph possesses a Grundy function.
This follows from Theorem 3.
Corollary 3. A graph without odd circuits possesses a Grundy function.
This follows from Theorem 5.

314 ORAPHS
Theorem 7 (Grundy [1939]). A graph G without circuits possesses a unit*,.
Grundy function g(x). Moreover, for each vertex x, g(x) does not exceed if,,
length of the longest path from x.
As in Theorem 4, consider the sets:
X{O)-{xfxiX,r(x)-0}
X{\)-(x/x$X{0)tr(x)<zX{0)}
XB)-\xlxt X@) u X(\), r(x) <= *@)u X(\)}
etc.
Clearly, these sets partition X, andxeX(k) if, and only if, the longest path
from x has length k. The values g{x) can be successively defined on the
sets *@). X(\). etc.
If xe X@), let g(x) - 0. If x* X(\), let g(x) - I.
If for each y e X(k), the value g{y) is uniquely defined and satisfies g{y) <
k, then for xe X(k + I), the value#(*) is uniquely defined and g(x) < k + I.
Q.E.D.
Before demonstrating the fundamental properties of Grundy functions, it
is necessary to define the cartesian sum of I-graphs:
Let C, - (*,. r,). Ca - (Xa, fa) Gt - (Xp, fp) be l-graphs. and let
P - { I, 2 p}. The normal product of these l-graphs is defined to the
I-graph G - C,. C Gp with vertex set
X - *, x X2 x - x Xp - fl *t
/«r
and with correspondence
*i ^r)- U
<«? J»p-t
|/|#0
The cartesian sum of these graphs is defined to be the I-graph C
C, + Ca + ••• + Gp wiih vertex set X - I Lf p A'i and with correspondence
.*a V U ({'i}x •" x {jf;
i.r
Finally, the cartesian product of these graphs is defined to be the I-graph
G - C, x Cj x ••• x Cr
with vertex set X - flitp ^i and with correspondence
i,xj xp) -

KERNELS AND ORUNDY FUNCTIONS
315
H
Fin. 14.6
Example. Consider two machines, and let Xx denote the set of possible
states for the first machine. For *,, xj e Xit let xi 6 /\(Xi) if state xi can
follow state *|. Thus, the first machine defines a graph Gx - (Xi.Fi).
Similarly, the second machine defines a graph Ga - (X2, Fa).
Let (xi. x3) describe the state x{ of the first machine and the state x3 of
the second machine. If the operator works both machines simultaneously,
Rraph Gv x Ca represents the possible changes of situations. If the operator
*orks only one machine at a time, graph d + Ga represents the possible
changes of situations. If the operator can work either one machine or two
machines simultaneously, graph Gi-Ga represents the possible changes of
situations.
Remark. The connectivity of a cartesian product of 1-graphs has been
"udicd by McAndrcws [1963], who has shown the following result:
If G and H are two strongly connected graphs respectively with the sets
and M(H) of circuits, then the cartesian product G x H ix strongly

316 0RAPH3
connected if, and only If, the lengths of the circuits in M(G)V M{H) are rp/0.
lively prime.
Similar properties for the cartesian sum have been studied by Aberth
A964], who has shown that:
Graph G + II is strongly connected if, and only if, both G and II are strongly
connected.
Other properties ofG+H and G x II have been studied by Picard [I97Q1
and Pultr [1970].
Proposition 1. If 1 -graph G ■ (X, F) possesses a kernel S, and if \-graph
//- ( Y, A) possesses a kernelT, then the normal product G. II has the cartesian
product S x T as a kernel.
It is left to the reader to verify the stability and absorption of set 5x7".
Does the cartesian product of two graphs, each with a kernel, also have a
kernel?The example in l-ig. 14.6. due to C. Y. Chao[1963]. provides a counter
example: graphs G and // respectively have kernels {d} and {y), but graph
G + II does not possess a kernel, because such a kernel would necessarily
contain vertex dy, (since F(d, y) - 0) and two of the three vertices ax, hx, or
ex, which contradicts stability.
Later, it will be shown that if both graphs G and // possess a Grundy func-
function, then graph G + II has a kernel. Before demonstrating these results some
preliminary developments are required.
The binary expansion of an integer p is a sequence {p\pa, ...,p") of digits
such that
p -p* + 2p2 + 4/>J + ■■■ + 2*"V-
Its binary Jorm is
lor
cxnmplc:
Decimal Binary
form form
0
1
2
3
4
5
0 -
1 -
- 10 -
- II -
- 100 -
- 101 -
Binary
expansion
@)
A)
@.1)
(i. n
@.0.1)
A.0.1)
'...,V.
Decimal
form
6 -
7 -
8 -
9 -
10 -
II -
Binary
form
110 -
III -
1000 -
1001 -
1010 -
1011 -
Binary
expansion
@. iTl)
(l.l.D
@.0.0. 1)
(I.O.O.D
@. 1.0. 1)
A.I.O.D

KERNELS AND ORUNDY FUNCTIONS 317
Let (njo) denote the integer n modulo 2 (which is the remainder of « divided
by 2).
The digital sum of the integers />i,/>a />„ is defined to b: an integer
p ' Pi + P* + •■• + P* obtained from the binary expansions of the p, by
letting:
'-([i*1] .[trf] .[ift3] .»■)■
\ll-l JB) ll-l JB) l(-| JA) /
For example, we obtain:
3 + 7 -0,1) + A,1.1)- @.0. I)- 4
I +3 + II - (I)+ 0,1) + 0.1,0. 1)-A.0,0.1)-9, etc.
Proposition 2. The digital sum has the following properties:
A) Associativity: {p + q) '+ r ■ p '+ (q + r) ,
B) Existence of an element 0 satisfying: p + 0 - /;.
C) For «k7j />, existence of an element (— /;) satisfying: p + (—/>) - 0.
D) Commutativity: p + q - q + p.
The proof follows immediately.
Thus, the first three properties imply that the integers under the binary
sum operation form a group. The last property shows that this group is
abelian.
Corollary. Given two integersp andq, there exists a unique integer r such that
p + r -q.
This is a well known property of groups.
Theorem 8. If the \-graphs Gi - (Xlt f),.... <7B - {Xn, fn) respectively
possess Grundy functions gx(x), ...,gH(x), then the function
'* a Grundy function for the cartesian sum G - Gi + G8 + ■■■ + GH.
Consider a vertex (*t, x* xn) of G, and let
ft(*i) - Pi - (Pi'.P?.-) (' - <.2 «)
g(*i *«) ■ Pi + Pa + - + pB - p -(p'.p2,...)
where

318 ORAPHS
I. We shall show that for each q < p, there is a vertex
0"!. yit ....>"«) e H*i. *i xj
such that g{yx, ya yn) - q.
Ut *0 be the largest k for which p" q". Then, /?*« ■ I, q*« ■ 0, because
otherwise, we would have
0+ Y 2*-V>2*<'-1+ T 2*"V
*<*o k</k0
and
which is impossible.
Thus.
i-i
Without loss of generality, suppose that rf« - I. From the preceding
corollary, there is an integer r such that p + r - q, and. consequently,
Pi + r < Pi .
Hence, there is a vertex .>>, /"^.v,) in graph Cx with gi(yi) - Pi + r. Hence
2. We shall show that if (>>,, xa xB) 6 f(x,, xa xB), then
p.
If >>, 6 r,(x,), then giO'i)-# gi(Ai). Therefore, if we had
g2(x2) +
then, from the preceding corollary. ^r,(>',) ■ jfiCvO. which is a contradiction.
Therefore, g is a Grundy function for C.
Q.E.D.
4. Nim games
Given two players A and B and a I-graph (X. F), the following game can
be defined: Starting from some initial vertex x0. player A selects a vertex *i
from r(x0). Player B selects any vertex xa from /'(*i). Next, player A selects
any vertex x3 e r(xa), etc. If a player selects a vertex x, with f(x,) ■ 0. then

KERNELS AND ORUNDY FUNCTIONS
319
that player wins, and his opponent loses. Clearly, if there are circuits in the
graph, the game need not terminate.
This game is called a Nim-type game. We shall study characterizations or its
winning positions, i.e., those vertices which must be chosen by a player in
order to win no matter how his opponent plays.
Example I. Fan Tan. There are p piles of matches. Each of two players
alternately select a pile and remove one or more mutches from it. The player
who removes the last match wins.
Example 2. Two players play with the same rules, but the last player to
remove matches is the loser. This game was popularized by the film "Last
Year at Muricnbad" by Alain Resnais; it reduces to a Nim game by adding
one vertex to the graph.
Example 3. Each of two players alternately place a tile that covers three
squares in a straight line on an n x n chessboard. No square can be covered
twice. The last player to place a tile wins. ■
If/i is odd, the first player can win by placing his first tile in the centre or
the chessboard. Then, he should place his second tile in a position that is
symmetric with respect to the centre or the chessboard to the first tile placed
by his opponent, etc. (This is always possible since the centre or the chess-
chessboard has been covered.) Thus, his opponent must lose.
Example 4 (Withoff [1907]). Let X be the set of non-negative integral
points in the plane. From an integral point (/>, q) in X, a player can select any
9 10
Fig. 14.7

320 ORAPIIS
point such that the value of one of the co-ordinates decreases and the value of
the other co-ordinate remains unchanged, or such that the value of both co.
ordinates decrease by an equal amount. The players take alternate turns. The
first player to select the origin @,0) wins.
For example, from point D, I), the following choices arc available: D 0)
C, I). B. I). (I. I). @,1), C,0). ' *
The graph (X, F) defined by this game contains no circuits, and, conse-
quently, it possesses a Grundy function, indicated in Fig. 14.7. The winning
positions arc circled. Note that these positions are distributed symmetrically
around the main diagonal.
The above game is due to R. Isaacs [1958], but it was noticed by J. Kcnyon
[1967] that it is equivalent to a game invented by WithofTin 1907.
Theorem 9. If the graph (X. f) possesses a kernel S, and if a player chooses
a renex in S, this choice assures him of a win or a draw.
If player A chooses a vertex x, in S. cither f(X|) ■ 0 (victory), or his
opponent B will be forced to choose a vertex xa in X — S. Then, player A can
choose again a vertex x3 in S, etc. The game ends when one of the players
chooses a vertex xk with f(.vfc) » 0. Clearly, xk 6 S, and the winning player
cannot be B.
Q.E.D.
To win games of this form, a player could calculate a Grundy function g(x)
(if there exists one) and then play on the kernel
S-{x\g(x)-0).
If the initial vertex x0/satisfies g(x0) - 0, then player A is in a dangerous
position because his opponent can get a win or a draw. If g(.v0) ** 0, then
player A can ussure himself of a win or a draw by choosing any successor Xi
of x0 with g(xt) ■ 0.
Consider n Nim games (*,, T,), (X3. Ta) (XH, rn). Suppose now that
each player can play only one of the Nim games during his turn; the first
player who cannot play at all is the loser.
This situation is in fact a Nim game on the cartesian sum of the graphs
(A",, r,). The following theorem gives a winning strategy.
Theorem 10. Consider n Hint-games d - (*,. T,), Ga - (Xa, r2)
C ■ (Xn, r,) with Grundy functions gltga gn. A player will not lose the
game on the cartesian sum C ■ 2 G, if he chooses a position
x ■■ (Xi, X2,.... xn)

KERNELS AND OKUNDY FUNCTIONS 321
that
gi(*i) + gj(*j) + ••• + gn{xm) - 0.
The proof follows immediately from Theorem 8 and 1 hcorcm 9.
Example. Fan Tan. Each of two players alternately selects one of p piles
of matches and removes at least one match from it. The player who removes
the last match wins.
This game is the cartesian sum of the games (.Vi, /*,), (Xa, f2),..., {Xn, f,),
where xh e Xk represents the state of the Ar-th pile. Clearly gk(xk) equals the
number of matches in the Ar-th pile.
From Theorem 10, a position is winning if, and only if, the digital sum of the
number of matches in the piles is zero.
Consider a Nim game G ■ (A", O without circuits and with a Grundy
function g(x). Let + be any operation that associates a vertex z to every
ordered pair of vertices {x, y). This is written as z ■■ x + y. For S <= X and
T <= X, let
S + 7"- {s + tfseS, teT}.
The following result produces a winning strategy:
Theorem 11 (Grundy [19391). Given a \-graph C - (X, O without circuits
ami an operation + such that
r(x + y) - (r(x) + y) u (x + r(y)),
then the Grundy function gofG satisfies for all x,yeX,
0) g(x + y) - g(x) + g(y).
We shall prove the theorem by induction. Consider the sets
-{*//"(*)-0}
X0)-{x/r{x)<zX@)}
XB)-{xinx)<zX{\)},clc...
Since the graph contains no circuits, X - X(p) for some value of p. Fur-
Furthermore, we have:
* + >>€X@) m. (r(x) + y) u (x + r(y)) - 0 * x,yeX@).
Thus, x + ye X@), implies that:
g(x + y) - 0 - 0 + 0 - g(x) + g(y).

322 ORAFIM
Therefore, equation (I) is satisfied on A"@).
Suppose that equation (I) is sntisfied on X{k - I); we shall show that it j,
satisfied for a vertex z - x + y in X(k).
If this were not true, then cither
Six) + g(y)
or
g(z) < g(x) + g(y) •
Case I. Suppose g{z) > g(x) + g(y). There is a vertex r, e F(z) such that
- g(x) + g(y) •
Without loss of generality, we may write zx - x, + y, x, e F(x). Since,
2, 6 T(r) c A"(A - I), we have \
Hence,
g(*) + g(y) - K(*i)
Therefore, #(.v) - R(xx). which contradicts that x, e T(x).
Casf 2. Suppose j?(z) < /f(x) + g(y). Then, as in Theorem 8, there is an
integer r such that
g(z) - r + g(y), r < g(x).
Since there is a vertex xTe F(x) with g(.v,) ■ r, then
g(z) -g(x,) +
Since X! + ye F(:) c *(* - |), we have
which contradicts that *, 4- y e T(z).
Thus, in both cases, we have a contradiction.
Q.E.D.
Corollary. Z,ef C - (*, f) Ae a \-graph without circuits, and let + A* fl"
operation on X such that
(\) x + y y + x,
B) x +(y + z)-(x + y) + z,

KLRNBU AND ORUNDY FUNCTIONS 323
0) x + r - >> + z •*■ x - y,
i4) there exists a vertex e 6 X such that x + e - x for all xe X and such that
r(e)-0.
(j) A* + y) - (rw 4- >) u(x
A vertex z is defined to be "Irreducible" if there exist no vertices x and y
distinct from e such that
z ■* + )'•
Each vertex xe X can be uniquely written as a digital sum of irreducible
vertices. If the value of the Grundy function g{z) is known for all the
irreducible vertices, then g(x) is defined for x by
g{x) m g(z, + z2 + -) - g(z() + g(z2) + - •
The uniqueness of this decomposition is a well known result from the theory
of semi-groups.
Example. Two players alternately choose one of n piles of matches and
divide the chosen pile into two unequal piles. The last player who can make
such a division is the winner.
A position x is represented by the numbers XltXa xk that denote the
numbers of matches in each pile. Let
* + y -(*i,Xa....,*») + (Ji, J2.-.Pi) - (*i.*a.. ■-.**• >'i yd-
The operation + satisfies:
A) x + y m y + x,
B) x + (y + z) - (x + y) + z,
C) x + z - y + z ^ x ■ y,
D) There exists a position e (no matches) such that x + e - x for all x and
&uch that r(e) - 0,
E) r(x + y) - (r(x) + y) o (x + r(y)).
Thus, the value of the Grundy function g for the irreducible positions per-
permits the calculation of the other values by means of a simple digital sum. An
irreducible position corresponds to a single pile with Xx matches, and some
values of gEtl) are given below

324
GRAPHS
*l -
g(Xi) -
1
0
2
0
3
1
4
0
5
2
6
1
7
0
8
2
9
1
10
0
II
2
12
1
13
3
14
2
15
1
16
3
17
2
18
4
19
3
20
EXERCISES
1. Let A be an integer such lhai
max { n — m, I } < X < (n + I —
Show (hnt I here exists a simple graph C with « vertices and m edges such thai Ihc cor-
corresponding directed symmetric graph G* wtisllcs /}((#•) - A.
2. Show that in the Nim game in Example 4. Ihc abscissa of the /r-tli winning position
below ihe diagonal is
3. Show that for the Muricnbud game (Example 2) there Is a Grundy function g'lx) (hit
can be constructed from the Grundy function for Fan Tan (Fxamplc I) as follows:
g'(x) ■ itix) if in position x, there is at least one pile with more than one match
g'(x) ■ I - jrt.v) if no pile has more ihan one match.
4. Two players arc presented with n piles of matches. Let / < n. Each player must alter-
alternately choose q piles and remove ut least one match from each of Ihc chosen piles. The
last player to remove matches wins.
In order lo determine if ti position is winning. It is necessary lo calculate in binary (he
number (pi,, pi, ...) of matches in the A-ih pile und to show that for each integer r,
- It/'} -»■
where Mvi1*) denote* (he integer remainder resulting from the division of A by q t I-
Docs this rule define a Grundy function.' (F. H. Moore 119091)
5. Show that if Cu G G, are l-graphs respectively with pscudo-Grundy functions
guga St. their normal product Gx Ga G, possesses a pseudo-Grundy func-
function
ir(*i, jtz. .... Xp) — max^((.t()
f«p
(Butsan. Varvak JI9MJ)
6. Lei G and // be I-graphs. Show that (heir cartesian sum G + H contains an culcrwn
circuit if. and only if. both G and H contain culcrian circuits. (Abcrth |I964)>
7. I el G be a graph that is not strongly connected. Let p be Ihc cardinality of a kernel
of graph G, ihc transitive closure of G. Let grnph // be obtained from C by reversing the
direction of each arc. Lci q be the cardinality of a kernel of graph //. Show that the mini-
minimum number of arcs that must be added lo 0 to make a strongly connected graPn
equals max \p,q).
Him: Use Exercise 14, Chapter 7. (D. Roy I

CHAPTER 15
Chromatic Number
I. Vertex colourings
ftot^tvomatic number of a graph G is defined to be the smallest number or
flours needed to colour I he vertices or the graph so that no two adjacent
vertices have the same colour. The chromatic number or graph G is denoted
by -AC).
A graph G with y(G) < k is called k-cohurable. A k-colouring of the vertices
u a partition (Si, Sa Sk) of the vertex set into k stable sets, each repre-
representing the vertices of one of the k different colours.
In (his chapter, we shall assume that graph G is simple.
Example I. The cfiromatic index (see Chapter 12) of G is equal to the
chromatic number of a graph L(G) formed in the following way: The vertices
of I(G) represent the edges of G, and two vertices are adjacent in L(G) if,
and only if, the corresponding edges in G are adjacent. Thus, results for the
chromatic number yield results for the chromatic index.
Example 2. Is it possible to colour all the countries on a geographic map
with four colours so that no two countries with a common border have the
same colour? This famous problem, called the "four colour problem" has
not yet been solved.
A geographic map corresponds to a planar multigraph G whose edges are
ihc borders between countries. Colouring the countries corresponds to
colouring the vertices of the topological dual G* (see Ch. 2, §2). Thus, the
■nap colouring problem is equivalent to showing (hat all planar graphs are
Colourable.
Example 3. Let X represent the inhabitants of an aquarium. Consider the
''graph G - (X, /") with tl5(x) < 2 for all x e X. Is it possible that this graph
^Presents a family tree, i.e., y e /"(*) implies that y is the child of xl Clearly, a
necessary condition is that the graph contains no circuits and its vertices
^n be coloured with two colours «t (male) and aa (female) such that rj x)
c°ntains at most one clement of each colour, for each x.
325

326 oraphs
Fid. 15.1. Geographic map corresponding 10 a 4-colourable graph G. The colours
0. I. 2. 3 correspond lo the values of u Grundy function for ihc symmetric (directed)
graph C
It is also necessary that the simple graph // - (X, £), where [x,y]e £if,
and only if,
rx n /> * 0 ,
is bicolourablc. "*
Example 4. Examination schedule. At the end of the term, each student
must take an examination in each of his courses. Let X be the set of different
courses and let Y be the set of students. Since the examination is written, it is
convenient that all students in a course take the examination at the same time.
What is the minimum number of examination periods needed?
For each examination x, let S(x) c y denote the set of students who must
write examination x. Form graph C « (X, E) where [x,x']eE if S(x)^
S(x') ft 0, i.e., if examinations x and x' cannot be given simultaneously-
Each vertex colouring of this graph corresponds to a possible examination
schedule, and vice versa. Thus, the examination scheduling problem >s
solved by finding y(C).
Example 5. The schoolgirl problem (Kirkman). This famous problem c*n
be stated as follows: Each day the fifteen schoolgirls a, b, c m, n, o from 8

CHROMATIC NUMBER
327
larding school take a walk in 5 rows of 3. Is it possible for them to take 7
.alks such that no two girls walk together more than once?
j)y considering symmetry, Caylcy found the following solution:
Sunday
afk
cnm
din
ejo
Monday
abe
cno
dfl
ghk
ijm
Tuesday
aim
bcf
deh
gio
jkn
Wednesday
ado
bik
cjl
Thursday
agn
bdj
cek
fmo
hii
Friday
ahj
bmn
cdg
efi
klo
Saturday
ad
bho
dkm
eh
fgj
The schoolgirl problem is related to the well known Steiner problem: Is it
possible for the 15 schoolgirls to form successively 35 distinct triples such
that no two girls appear together twice in a triple? To solve this problem,
form a graph G whose vertices arc the j I - 455 possible triangles. Two
vertices arc joined together if the corresponding triangles have two girls in
common. Thus, the problem reduces to finding a maximum stable set 5.
Clearly, | S | < 35, because no girl can appear in more than 7 distinct
triples of S, and therefore there are at most 15 x 7 x i ■ 35 triangles).
If a stable set S has 35 vertices, then 5 is a maximum stable set.
To check if a solution 5 of the Stcincr problem yields a solution of the
schoolgirl problem, construct a graph C whose vertices arc the 35 selected
iriplcs, and with two vertices of C triples joined by an edge if the corres-
corresponding triples have one girl in common. If the chromatic number y(G') - 7,
'he schoolgirl problem is solved. If y(C') > 7, it is necessary to select another
A set of triples which meets the conditions of the Stcincr problem is also
called a Steiner triple system of order 15, or a A5, 3, \)-block design. It was
proved by the Reverend T. J. Kirkman. in 1847, that a Steiner triple system of
order n exists if and only if n m I or 3 (mod 6).
Now, if n - 6 k + 3 girls take a walk each day in 2 k + 1 rows of three,
c&n an arrangement be made for 3 k + I different days such that any pair of
8irls belong to the same row on exactly one day of the 3 k + I days? The
c*istcncc of such an arrangement for all A > 0 was proved by Ruy-Chaudhuri
and Wilson. [1971).

328
ORAPHS
Algorithms to determine chromatic number. Several algorithms exist
the determination of a minimum vertex colouring of a graph. The
efficient of these algorithms use the following two principles:
I. Principle of contraction-connection.
Consider a simple graph G with two non-adjacent vertices a and b. "Tv
connection CI ab of graph G is obtained by joining two non-adjacent vertices
a and b with an edge. The contraction C:ab of graph G is obtained by
shrinking set {a, b) into a single vertex c(a. b) and by joining it to each
neighbour in G of vertex a and of vertex b.
C
G/ab
G:ab
a. (I
b it
e.h a.tJ
P)
a.d
e.b
15.2

CHROMATIC NUMBER 329
colouring of G in which a and b have the same colour yields a colouring
0{-ab- A colouring of G in which a and b have different colours yields a
flouring of C / ab.
Repeat the operations of connection and contraction, until the resulting
graphs are all cliques. It' the smallest resulting clique is a A-clique, then
o - *•
For example, the graph G in Fig. 15.2 yields via the connection-contraction
process a 3-clique. A 3-colouring of the original graph is obtained by recover-
recovering the original from the 3-clique. Vertices a and </ will be coloured «; vertices
t and I) will be coloured /?; vertex c will be coloured y.
II. Principle of separation into pieces
Suppose that during the above contraction-connection process, we en-
encounter a graph with an articulation set A that is a clique; the removal of/f
creates connected components C1( C3, C3 and, clearly, the subgraphs
G^ci - 'Aa. GavCi - "a. GAuC9 - . ••• can be coloured separately so
that the colours of the vertices of A do not conflict.
Hr. 15.3
For example, graph G in Fig. 15.3 is 3-colourable because each subgraph
i is 3-colourable.
piece of graph G relative to the articulation set A is defined to be a sub-
of C of the form //, - GAvCl. Clearly, the pieces of a graph can
coloured separately.

330 GRAPHS
Theorem 1. If G is a graph of order n, then
y(G)«(G) > n,
y(G) + a(G) < n + I .
Let us colour the vertices of G with q - y(G) colours a,, a a,.
denote the set of vertices with colour a,. Since S, is a stable set, we have
n - | S, u S2 u - u Sf | - £ | S, |
i-i
Furthermore, if a maximum stable set 5 is coloured with the first colour, then
n — st(G) colours can be used to colour X — S. Hence,
y(G) < (« - )
Q.E.D.
Ihcorcm 2 (Gaddum, Nordhaus [I960]). Let G be the complementary graph
of a simple graph G. Then
y(G)
Recall that the complementary graph of G — (X, E) is defined to be G*
(X, 2(X) — E). Clearly, the theorem is true for n a I and // - 2.
We shall suppose that the theorem is valid for every graph with n
vertices, and we shall show that the theorem holds for a graph G with n
vertices, n > 2. Let Go be the subgraph obtained by the removal of vertex
x0. Clearly, we have
@ y(G) < y(G0) + l.
B) y(G) < y(G0) + 1 .
1. If in both (I) and~f2) equality holds, then
y(G0)
v(G0).
Hence,
y(G0) + y(G0) < da(x0) + da(x0) - n - I .
Hence
y(G) + y(G) < (« - 1) + 2 - n + 1.
2. If either (I) or B) is not satisfied with equality, then, by the induction
hypothesis,
y(G) + y(G) < y(G0) + y(G0) +!<« + !.

CHROMATIC NUMBER 331
Hence, the theorem is valid in both cases.'"
Q.E.D.
Remark. This bound is the best possible: A graph of type Ti(n,p) is defined
to be a graph with n vertices formed from a stable set 5r with p vertices and a
clique Kn.p*\ with n - p + 1 vertices such that
I
1
1
F(8. 15.4
If graph G is of type 7",, then
y(G) - n - p + 1,
Hence y(G) + y(G) - n + 1 .
Similarly, a graph of type T^n,p) is defined to be a graph with n vertices
formed from a cycle Co of length 5 without chords, a stable set Sp with
P < n - 5 vertices, and a («-/>- 5)-clique A'n.p-C, such that these three sets
arc disjoint and
A) each vertex C6 is adjacent to each vertex of A'n-p-o.
B) no vertex of C6 is adjacent to a vertex of Sp.
If G is a graph of type T3, then
"' A Mronjcr inequality has been obtained by K. P. Gupta [1968].
*>quasl-colourlnx wlih k colours is defined to be a mapping/: X — (ax, ag. •••. oti.) well lhat
' * I Impliei the existence of two adjacent vertices x and y with f(x) ■ a, and fiy) ■ a,. A
Wounng with a minimum number of colours it a <iuuti-colouring because, otherwise, two colours
•< tnd o, arc not udjaccm und the vcriice* with uny one or llicse two colour* could be coloured
°y uwng only one eolour.
1 e> <K(i) denote ihc maximum number or colours needed Tor a quasi-colouring or C. Thus.
tfG) . Oupiu h«t shown that
AC) + v(C) < » + 1
Clearly, this result Is stronger than Theorem 2.

332
Hence
GRAPHS
y(G) -(r»-p-5) + 3-r»-p-2.
y(G) ■ p + 3 .
+ v(G) - n + 1.
C« 7,A1,4)
Flu. 15.5
The bound given in Theorem 2 is attained by graphs of type 7\ and 7",.
Furthermore, H. J. Finck [1966] has shown that only the graphs of types Tx
and Ta attain this bound.
Corollary. IfGis the complementary graph of a simple graph G ofordirn,
then
By Theorem 2, we have
- y{G)J
(y(G)
(n
Q.E.D.
Remark. This bound is the best possible. If/i is odd, then each graph Go(
type
or
satisfies
If /i is even, then each G of type

CHROMATIC NUMBER
333
satisfies
1 hcorcm 3. If G Is a simple graph with n vertices and m edges, then
tr - 2 m
Let y(G) " q. Let Si, 53 54 denote the sets of vertices of colour a,,
a a,, respectively. The adjacency matrix of graph G has the form:
0
Hi
0
•
0
Let n, - | 5, |. Let No denote the number of 0 entries in this matrix,
and let W, denote the number of I entries in this matrix. Then,
No
2m,
(«i + "a + — +
n1
-
T''is follows from the Cauchy-Schwartz inequality | x | | y \ > \ < x. y > |,
wi)hx - («,.«, it,) and y-(I, I I).
I'hus, the total number of entries in the matrix satisfies:
n1
Hence,
f,>2m + -.
<7
2 - 2 m) q > n2.

334 GRAPHS
Hence the required formula.
Q-E.D.
n
Remark. In order to have equality, it is necessary that No - ^ . and that
which implies that (nltn , «,) is proportional to A,1 1). Thus,
equality holds only if
"i m ni m '" m "f i
and if the matrix has only I entries outside of the blocks S, x St, i.e., the
graph is formed from q stable sets 5t, Sa S, of the same cardinality
with two vertices joined together if, and only if. they belong to distinct S,
Theorem 4. (Roy [1967], Gallai [1968]). If G - (X, E) is a simple gi'aph
with y{C) ■ q, then, for each orientation oftts edges, there exists an elementary
path with q vertices. Furthermore, there exists an orientation for which there is no
path with more than q vertices.
1. Suppose that G - (X, O is a I-graph, Let i{fi) denote the number of
vertices encountered by an elementary path ft (i.e., the length of path \i plus
one).
We shall show that max,, i(/i) > y(G) where the maximum is taken overall
elementary paths /< of G. More precisely, we shall show that the vertices ofC
can he coloured with maxu t{fi) colours.
By Corollary 2 to Theorem C, Ch. 14), there exists a set S? such thut each
vertex of G can be reached by a puth starting from 5?, and no two vertices of
S° arc connected by a path. Let
S°3 - ns0,) - S?;
sS - nsS) - (s? u sp
Clearly, these sets partition X.
Let {X, A) denote the partial graph generated by the arcs (.v,;') of G such
that
If. for some index /, there exist two vertices aeSt and b e S, with b e
then define a partition (.?[, Si, Si,...) of X, where

CHROMATIC NUMBER 335
S) -Sj f/ — 1.2 /-I)
Repeat the process until all sets in the partition arc stable.
The initial partition (S?. S$. SJ,...) has the property that for each p and
for each xeS%, there extsis an elementary path ftx terminating at x with
tll'x) - P- AH the successive partitions have this property because if (S*. SJ,
SJ,...) satisfies the property, then (S\tl, Sg + I, SJ*1,...) also maintains this
property for all vertices that do not change class (and also for the other
vertices, obviously).
Thus, a partition into h stable sets with the desired property can be ob-
obtained. Hence.
y(G) < h < max i(/0 .
2. This result is the best possible: let the vertices of a simple graph G be
coloured with q m y{G) colours, a,.aa a,, and let each edge [x,y] be
directed from x to y ifx is coloured a, and y is coloured a, with / < j; then it
is evident that no puth of the resulting graph contains more than q vertices.
Q.E.D.
Corollary. Let G be a simple graph coloured with q - y(G) colours cclt«3,
•...a,. Then there exists an elementary chain that encounters consecutively
ifie q colours «, in this order.
Direct edge [x, y] from x to y if x is coloured a,, y is coloured with a,,
and / < j. From Theorem 4. there exists an elementary path containing
exactly y(G) vertices. Clearly, this path has the required property.
Q.E.D.
It follows from Theorem 4 that a complete graph contains a hamiltonian
Path.
Ihcorcm 5. Let {Sl% Sa,.. , SQ) be a q-colouring (not necessarily minimum)
°f a simple graph G, and let
dk - max dc{x).
xmSk

336 GRAPHS
Then,
y(G) < max min { it, dk + 1 }.
1. If 5, is not a maximal stable set. add vertices to S to obtain a maximal
stable set S[. If Su — SJ is not a maximal stable set in X — S[, then add
vertices to it to form a maximal stable set 5£ in X — S[, etc. This process
defines a new colouring
(S| i Sj, ..•, Sr),
with
mln (k,r I
U S't = Sk (k -1,2 q).
2. Let /(a) denote the index / such that x e S',. Let a-0 be a vertex with
i{x0) - k; since x0 is adjacent to each S', with / < k — I (by the maximally
of S;). then
^c(^o) >k-\.
Thus, for all vertices x.
3. Let xoe Sk. From Part I. it follows that i(x0) < it and. consequently.
i(x0) < max J(x) < max (cfc(x) + I) - dk + 1 .
xi Si, xi Si,
Hence.
y(G) < max i(x0) < max min {k,dk + 1 }.
Q.E.D.
Corollary I. Z.ef G be a simple graph. If for some integer q, the number of
vertices with degree > q Is < q, then G Is q-colourable.
Assume that the vertices x, arc indexed with decreasing degrees.
Consider the n-colouring ({x,}, {xa) {xk} {xn}). Thus, dk -
do(xk). If it < q, we have
min { dk + I, k } < q.
If k > q + I, we have
min {dk + I, * } < min {</,., + I, k } < min {<y, it} < 9
Thus, from Theorem 5, y(G) < 9.
Q.E.D.

CHROMATIC NUMni H 337
Corollary 2. A simple graph with maximum degree h. is (h + I )-colourahh>.
The proof follows by letting q - h + I in Corollary I.
Corollary 2 can be improved by the following result, known as the '* Brooks
Theorem"; the proof presented here is due to Mclnikov and Vizing [1969].
Ihcorcm 6 (Brooks [1941]). Let G be a connected simple graph with maxi-
maximum degree h. Then G is h-colourable, unless
(|) h 2, and G is a (h + ])-clique. or
B) // - 2, and G is an odd cycle.
Clearly, the theorem is true for h - 0, I. 2. Let G - (X, E) be a simple
graph with maximum degree h > 3, with y(G) • h + I, and that contains no
(A + l)-clique. We shall show that this leads to a contradiction.
1. Since vertices that arc not essential to the above properties can be re*
moved, we may assume that G is minimal with respect to these properties.
Let .r0 X. The subgraph Go generated by X — { x0) contains no (A + I)-
cliqiic, and therefore y(G0) < h + I. Hence, y(C0) - h. This implies that
</g(*o) > h because, otherwise, one of the h colours used to colour Go could
be used to colour x0, which contradicts y(G) - h + I.
Thus, dc(Xo) m h. We may assume that, if )'i,y3 y* denote the h
vertices adjacent tox0, they are coloured with the colours ai, «a, ..., a,, respec-
respectively in a given h-colouring o/G0.
2. Let G{x,, a,) be the subgraph of Go generated by the vertices with colours
«i or a, in the /(-colouring of Co. Vertices y, andyt belong to the same connected
component ofG(a,, a,) because, otherwise, after interchanging colours a, and
ff/in the connected component containing y,, x0 could be coloured x,, which
contradicts y(G) • // + 1.
3. We shall show that the connected component ofG(u,, x,) that contains
>'■ and y, is an elementary chain n\yx,y,\ going from y, to y,.
Vertex y, is adjacent to only one vertex with colour a, (otherwise, since
4>0'i) < h. vertex y, could be recolourcd with a colour ak, k /,,/, and vertex
*o could be coloured «,).
Consider a chain in G(«,, a,) from y, to y,. We shall show that this chain is
Uniquc. Let x be the first vertex of this chain with dmax, «,>(*) > 2. If x has
colour a,, for example, then there arc three vertices with colour a, adjacent to
*'< &incc t/0(.r) < h, vertex x can be recolourcd «k, k ij. This would dis-
disconnect y, ind y, and contradict Part 2.
4- We shall show that the two chains it[yt,yi\ andn\yt.yk] that constitute
tOt»ponents G(ot,, a,) and G(u,,ak) cannot contain a common vertex z »* y,.

338 GRAPHS
If such a vertex z existed, then it would have colour a, and would h*
adjacent to two vertices with colour a, and to two vertices with colour»
Thus, h > d(£z) > 4, and there is a fourth colour. st a,, a,, «k to recolourT
This would disconnect y, and y, and contradict Part 2.
5. Since G does not contain any (A + l)-cliqucs, there exist in ro(*0) tw
non-adjacent vertices, say yt and yt. Consider the connected component
w[j>i»ya] of Cfat, «a) and the connected component n'[yi,ya] of G(a,,a \
Let x be the first vertex after y^ in chain n[yt, >3]. Since >>, and ya are non-
adjacent, x >>2. If colours «! and a3 are interchanged in the component of
C(«,, a3) that contains yx and y3, then vertex >, is rccoloured «3 and vertex>■,
is rccoloured a,. Furthermore, the new component //ia with colours a, and
aa that contains ya satisfies
"u 'III'.^2).
since from Part 4, chains n[yi,yi] and it'[yi,y3] have no common vertex
(except yi).
On the other hand, the component Ha3 with colours «a and «3 that con*
tains yi satisfies
Ha3 = n[yltx),
since .v has colour «a and is adjacent to yx. Hence, x is a vertex common to
the connected components //,a and //a3. This contradicts Part 4 and com-
plctes the proof.
Q.E.D.
The next section will present extensions of the Brooks theorem.
2. Y-Critical graphs
A simple graph G is defined to be y-crltical if for each vertex x0 the subgraph
Co geneirated by X — { x0 ] has chromatic number y(C0) < y(C).
Note that in this case, y(C0) - y(G) - 1 because if y(C) - q + 1 and
y(C0) < q, then Co can be coloured with q - I colours and vertex x0 can **
coloured with a q-lh colour, which contradicts that y(G) > q.
We shall now study the properties of y-critical graphs.
Property 1. A graph G with y(G) - q + I has a -(-critical subgraph w"*
y(G) - ? + I.
If G is not y-critical, there is a vertex x0 whose removal does not decrease
the chromatic number. If the subgraph Co generated by X - {x0) is n0
y-critical, then again there is a vertex x, whose removal from Go does no

CHROMATIC NUMHfR 339
.grease the chromatic number, etc. Eventually, this process locates a
j^riticul subgraph. Q.E.D.
property 2. If G is a simple y-criiical graph with y(G) - q + I. then
Ux) * 1for ali *•
If ,/f.(.Vo) < q, the subgraph Co generated by X - {.v0} can be coloured
tfjth / colours. At least one of these q colours is not adjacent to vertex .v;
thus, x can be coloured with this colour, which contradicts y(G) - q + I.
Q.F-.D.
property 3. A y-critical graph is connected.
The proof is obvious.
. Property 4. A y-critical graph contains no articulation set that Is a clique.
Let G be a y-critical graph with y{G) - q + I. and let A be a clique that is
nn articulation set of G. Since G is y-critical.
Ml-
Let fl|. B-it B. be the connected components of the subgraph generated
by X - A, and let B\ - fl, u A, Ba - B'a u /I,... be the corresponding
pieces. Since 0 is y-critical. each piece can be coloured with q colours. Thus,
Ccan be coloured with q colours, which contradicts y(f/) — q + I.
Q.E.D.
Property 5. A y-critical graph has no articulation points.
This follows from property 4 since an articulation point is a I-clique.-
Property 6. // G ix a y-critical graph with y(G) - q + I, and if A -
{o,b} is an articulation set ofG, then there are exactly two pieces B\ and B'a
ivlatke to this articulation set and
Clearly, a piece B' relative to A can be coloured with q colours. Three
c«scs must be considered :
A) B' cannot be coloured with q colours so that a und h have the same
colour,
B) B' cannot be coloured with q colours so that a and b have different
colours,
C) There is a colouring of B' in which a and h have the same colour, and
fter* is also a colouring of B' in which a and h have different colours.
Case C) cannot apply to a piece B' because the corresponding component
8can be removed without changing the chromatic number of G. If Case (I)

340
CRAPHS
applies to /?'. then >'(/?) - q because if y(B') - q — I, vertices a and h c«n ^
coloured with a g-th colour, which contriidicts that Case (I) applies t0 »
Similarly, if Case B) applies to B\ then y(B') - q.
Since y(C) ■ </ + I. there is a piece satisfying Case A) and a piece satisfy
ing Case B). Since G is y-critical. there is only one piece of each type.
QE.D.
B pieces)
C piece* or less)
B pieces)
E pieces or less)
O pieces or less) B pieces) E pieces or less)
Fig. 15.6. Vnrious types of ariiculalion sets in u y-critical graph
Property 7. If C is y-ciiticat with y(C) - q + 1 > 4, and if A - { a, b,()
Is an articulation set, then one of the foil wing cases occurs'.
I. GA contains only one edge and A has al most three pieces, each »'*"
chromatic number q.
II. 6", contains exactly two edges, and A has at most two pieces, each **'""
chromatic number q.
III. CA contains no edges, and A has at most five pieces, each with chromo"c
number q or q — I. t
We shall show only Case I. For example, suppose that a and b are the
adjacent vertices in A. If piece B' is coloured with colours 1, 2 q< t'ien
vertices a, b, c may be coloured 121. or 122. or 123. However, not all of th*'6

CHROMATIC NUMBI.R 341
colourings are possible for B' because G is y-critical; it follows that
B 1 (because if y(fi') < ? - !• 'hen using a </-th colour, all three of
idese colourings would be possible).
If two of these colourings arc possible for piece B[, then there is a piece B'2
for which only the third colouring is possible (and there arc only two pieces
fincc G is y-critical).
If no piece admits two of the three colourings, (hen there exist three pieces
g[, B'a, B'3 that respectively admit one of each of the three colourings.
Q.E.D.
Property 8. A y-critical graph G - (X, E) with y(C) - q + I cannot be
disconnected by the removal ofq—\ edges, i.e.,
mb(A,X-A)>q (A c X. A + 0, X).
We shall show that a contradiction results if there arc two nonempty sets
A and B of vertices G with
AvB-X, AnB-0, ma(A,B)<q -I .
Since G is y-criticnl, the subgraph GA is (/-colourable; consider a q-colouring
function for GA, i.e., a mapping/(w) from A into {1,2 q} such that
x,yeA, [x,y]eE - /(*)*/(>).
Let fl,,«3, ...,ar be the vertices of A adjacent to B; choose function /so
that
f(a,)<i (/-I. 2 r).
Let eltea ep denote the edges that join A and B. with indices chosen so
that
e. - [«i. x), e, - [aJt y], t <j -* s < t.
Let gl be a (/-colouring function for the subgraph GB. We shall .show that
ty a sequence of transformations, this ^-colouring of GB can become com-
compatible with the ^-colouring of CA, which contradicts y«7) - i\ + I. If edge
fi is of the form [«!, b,], define a ^-colouring ga(y) of Gh by
- gl(y) if g*iy) »* 2. i
- 2 if gl(y) - g'(l
™ g'(b|) if g'OO ■ 2 .
gJ(^)
Thus
consider e2 ™ [o*. b,]. and similarly, define a ^-colouring ga(y) by

342 GRAPHS
- g2(y) 'f g2iy) »* 3. g2(bj)
g*(y) -3 if g2(y)-g2(bj)
- g2(b,) if g2(y) - 3 .
Thus,
«3(A;)- 3 >f{ak).
This process can be continued, and since p - /r;c(/f, /?)<</- I, the last
edge e, gives a (/-colouring ^f"*1 with
Thus G can be coloured with (/ colours, which contradicts y(G) - r/ + J.
Q.Lb.
Consider a y-criticul graph G with y(G) - <? + I; from Property 2, graph
G satisfies
min </<;(*) > q .
xtX
To study the structure of G, we shall consider the set
W- {x/xeX. 4W-?}.
and study the structure of the subgraph Gw generated by M.
lemma 1. Let ft - [ao<ai.fla ak.lta0\he a cyik of Gy, andletfbe i
(f/ + I ^colouring of G with only vertex aQ having the (q + I )-st colour; then
there exists a(q + I )-colouring g of G with only v rtex a0 having the (q + I h '
colour such that:
I g(x) -/(x) (
Let «|, aa,..., a,,i be the colours in colouring/, wheref(a0) -
Sine* dda,) - </, each colour «,, «a, .... as appears exactly once in rc(W"
Hence, interchanging the colours of a0 and of ax gives a (q + l)-colourin8
with only vertex o, having the (q + l)-st colour. This procedure can be i*"

CHROMATIC NUMBER 343
e around cycle ft until vertex a0 has colour «,4l. This produces the
^quired colouring g.
Q.E.D.
Lemma 2. let ft — [ao,ai ak-i,a0] be a cycle of Gu with no chord
Incident to vertex a0; then /i is an odd cycle.
|. Since graph G is (q + l)-critical, there exists a (q + l)-colouring/ofG
with only vertex a0 huving the (q + l)-st colour «, + 1.
Since do(a0) - q, the set r(,(a0) {fli.flk_i} contains exactly q-1
vertices, and none or these vertices lies on >/. Each or these vertices take a
different colour in colouring/; let su «a,.... «,.a be these colours.
2. We shall show that only the colours a,_i and a,, arc assigned to vertices
fl,,fla. •••' ai-\ in colouring/.
If this is not true, then there exists in cycle ft a vertex a,, i > 0, with colour
a,.j <i — '.</• After repenting /- I times the operation of Lemma
I, a colouring g is obtained with only vertex a0 having colour a,+ 1 and with
But, then, the set ra("o) has two vertices with colour a, in colouring g.
Thus,a0 can be rccolourcd with a colour other than >:„.,, which contradicts
3. In colouring /, all vertices of cycle >i, except o0, have cither colour
•«-i or a,. Since vertices ax and ak.x belong to ro(a0)and cannot h«vc the
»amc colour, cycle /< is necessarily odd.
Q.E.D.
• hcorcm 7. Let G be a y-critical graph with y(G) - q + I; then r/«(.v) > ^/
/or «// _Vi u/w/ (W./; bitxk of fa subgraph GM generated by set M - { x f x e X,
«o(jf) - q } is either a clique or an odd cycle without chords.
'roin Lemma 2, each even cycle of graph GM has at least two chords,
•hus, from Theorem G, Ch. 9), GM has the required form.
Q.E.D.
Remark. If q equals
h - max rfc(Ac).
xtX

344 ORAPIU
the Brooks theorem follows from Theorem 7: Let C be a graph with
max do(x) - It and y(C) - h + I .
Consider the y-critical subgraph C of C with y(G') - h + 1 (which exist,
by Property 1). Since dr,.(x) > h (Property 2), and sinceda(x) < do(x) $ f,
graph C is regular of degree h. In other words, the vertex set X' of graph C
is equal to the set
Furthermore, C is connected and has no articulation points (Properties 3
and 5). Since Cm - C is a block. Theorem 7 shows that G' is either a (h + \y
clique or an odd cycle without chords.
Hence graph G has cither a connected component thut is a (h + l)-clique,
or, if h - 2, an odd cycle without chords.
The Brooks theorem follows.
The next theorem is also an extension of the Brooks theorem.
Lemma. Let G be a ycritical graph with y(G) - q + 1 >4.IfG contains a
clique C - { c t, ca cv) with q vertices such that
t (dG(c,) -<i)*q-3,
then in a ((-colouring of X — C. the sets
a, - ro{Cl) - c
contain a common colour.
Suppose that
Since G is y-crilical. then by Property 2, dc{cx) > q. Since d^,ci) - q
for at most q - 3 of the <•,, then
- q.
Thus | Ax | - I. Let a, be the colour of the unique vertex of Ax in the
(/-colouring g of X C. Suppose that there is a set AlB that docs not contain
colour Si. (If several such sets exist, take index i0 to be as small as possible-)
We shall now show that this is impossible, i.e., clique C can be coloured
with the q colours used for X - C, which will contradict y(G) - q + '■
I ct x,, «a a, be the q colours of g(X - C). Define the (/-colouring
j?(.v) of C successively in the following way:
— let #(<•,„) be colour a,;

CHROMATIC NUMBER 343
^.if q i0. let g(cv) be any colour different from the colours of A u
^-if q — I 'o» Id Jf(f»-i) be any colour different from the colours of
^•ctc...
Mote that, fory < q.
i() f
Thus.
or
Let #(c,) take a value in the set {a,, aa,..., a,} if the set
Ai^> {ci0» <■«•<■«-1 cy+i )
has less than q vertices. This is possible if
i.c. if
i.e.,
/ - (</ + 4)> + 4 </ < 0.
Since the roots of this quudratic equation are / - 4 and y" - q. vertex
f/can be coloured for:
It remains to colour c, torj ■ I. 2, 3. Since
rfo(c,) - f/c(c2) - da(c3) - ^.
** may take: /I, - {a, }. /l2 - { «2 }. ^j - { flj }•
lr 'o 9* 2, 3. then j?(er,) - g(aa) - ^@3) - a,. Moreover, C - { c,, ca, c0}
"as been coloured with ^ - 3 colours, including «|. Thus, r,, ct, c3 can be
floured with 3 colours diiTcrcnt from a,.
^ /0 ■ 3, then /?(<?,) - /?(«;,) - S|. Moreover, q - 3 colours have been

346 ORAPHS
used (including colour «i for c3). Thus, r, and ra can be coloured with 5
colours different from at.
If /0 — 2, a similar result follows.
In all cases, C can be coloured with the q colours that have iilrcndy bee
used for X - C.
Q.E.D.
Theorem 8 (Dirac [1952]). Let G be a graph with chromatic number
y(G) - q + I ami without any (q + \)-cliques. let S - { jc/*/,,-(*) > </}. 7/,w
- <?) > <? - 2 .
If q - I, the theorem is trivial because G cannot have both chromatic
number 2 and no 2-cliqucs.
If q - 2. then
Suppose that the theorem is true for all graphs with chromatic number
< q; we shall show that it is also true for a graph G with chromatic number
y(G) ■ q + I. where q 5> 3. (From the Brooks Theorem, we already know
that 5 0.) Suppose that G contains no (q + I )-cliqucs and that
I ( - </) < 9 - 2.
xtS
We shall show th.it this lends to a contradiction.
We may assume that G is y-criticil. (Otherwise, replace G by a y-critical
subgraph.) Since (lc(x) > q for all .ve X. then
(») - I (dcW -q)<q-2.
xmX
We shall now produce a contradiction for each of the following two possible
cases:
Case I. G contains no {-cliques.
Let .v0 be a vertex of S. (By the Brooks theorem, S contains at least one
vertex.) Let X - T be a maximal stable set that contains .v0, and let GT be
the subgraph generated by T. Clearly, y(Gr) < q + I (because 7" X and
G is y-critical).
If y{Gr) < q, then G could be coloured with only </ colours. Thus
y(Gr) - q.
Let
S'-{.v/.ver. dtt,(x)><i- I}.

CHROMATIC NUMBER 347
#e have
dc.r(x) + I <<U*) (xeT),
(because each vertex x s T is adjacent to X - T, since X - T is a maximal
stable set.)
Hence S' <= S.
In fact, 5' is a proper subset of 5 since .v0 € 5 and x0 i S'. Hence,
[dCr(x) -(</-!)]< I (</,,(*) - q) <
its'
I (o«) ?
Xf S
Rut, from the induction hypothesis, graph GT satisfies
</Cr(.v) - U/ - l)]>(fl- I)-2,
which is the required contradiction.
Case 2. C contains a q-clique.
Let C - {r,, ra,.... r,} be this clique. Set At - /(r,) - C is not empty
(because do(ci) > q since C is y-critical). Set A - T^C) C contains at
least two vertices (because G contains no (q + l)-cliqucs).
Renumbering the vertices if necessary, we may assume that
dtMi) < dcki) < - < '/<;(<•«) ■
Since </<,(<-,) - <; > 1 for at most q - 3 of the r,, from inequality (I), it
follows that
Thus, let o, be the unique vertex of Ax - rt;(ci) - C Since G contains no
Ffc. 15.7.

348 GRAPHS
(q + l)-cliques, there is a vertex c, that is non-adjacent to ax. (If several
such vertices exist, let y be the smallest possible index.) Let
Let C be the graph obtained from Gx- • by joining vertex a, to vertices
bi,b3,.... Since Gx.c is (/-colourable the graph C - { fli} is ^-colourable-
hence
y(G')<<7 + I.
Furthermore, from the lemma, graph C cannot be coloured with q colours.
Hence,
)'(G") - q + 1 .
Let G" be the y-critical subgraph of C with y(C) ■ q + I. Let graph H
be obtained from G" by removing the edges of the form (a,, /;,]. -^
Graph H contains vertex «, and one of the vertices of A,, because if C
did not contain an edge of the form [a,, />,]. then y(G") < y{Gx-c) < </■
Note that
<M«i) > da (fl,) - \A, | + (/ - 1) > q - \A, \ - 1 + j .
If | A, | > 2, we would have
fl] > t (dG(c,) -q)
7*{tl-j+ |)(|^|-|) + (-|^|- i+J)m
-(?-;+D(M;l-i)-(l^l-i) + i 2-
- u - »(M/1 - i) + y - 2 >
> ui -./)+;- 2 -</ - 2.
This contradicts inequality (I).
Hence. \A,\ - 1.
Let b be the unique vertex of A,, Graph // - ( Y, F) can be formed from
the (q + l)-critical graph G" by the removal of the only edge [fl|, b]. Thus,
''«(«i) - d,, (o,) - I > ^ - I
</,/(h) - do-(W - I > ^ - 1
</„(.*) - ^ (.veV. x n,,
Combining this with Property 8 gives

CHROMATIC NUMBER
349
- *) > I (do(x) - q) -
V.x - y) + 5; (</,,(x) - ,) > 9 - 2.
this contradicts inequality (I).
Q.E.D.
Corollary. Let G be a y-critical graph with y(G) ■ <y + I, m-//Aoh/ (<y + I)-
cllques, and with n vertices and m edges: then
2 m > (n + \)q-2.
Since da(x) > q for each vertex x, we have
2m- £ do(x) + (n-\S\)q>q-2 + \S\q + nq-\S\q-
xtS
-(/? + l)q- 2.
Q.E.D.
Remark. The inequalities of Theorem 8 and its corollary arc the best
possible Tor q > 3. To construct a graph G with
V(C) - q + I,
such that the inequality of Theorem 8 holds with equality.
C-
and
and remove edges
and add edge [c2. dq.,]. Sec Fig. 15.R.
FIr. 15.8.
■ It. « — 7, fl — 3

3S0
GRAPHS
Clearly. y(G) ■ q + 1, and C contains no (q + l)-cliquc. Furthcrrnoi*
is the only vertex of 5. and
- 2 </ - 2 .
Thus,
I
x«S
OE.D
3. The Hajos theorem
Let C be a simple graph. An elementary contraction on G is defined to
be any opcrution that rcplnccs two adjacent vertices a and b of C by a single
vertex c and joins c to each vertex of
If the contraction operation is repented enough times, a clique will be
obtained. Sec Fig. IS.9.
</<■«•
FIB. 15.9.
One of the most interesting conjectures concerning the chromatic number
of a connected graph is due to Hadwigcr:
Conjecture (Hudwigcr [ 1943]). Every connected graph G with y(C) - q «"' **
transformed into a q-clique by a sequence of elementary contractions.
For </ < 4, Diruc [19S2] and ILilin have validated the conjecture. For
q ■ 5. Wagner A964] has shown that Hadwigcr's conjecture is equivalent
to the four colour conjecture.
Let f „ be the class of all graphs with y(G) > q. Consider the
three operations in r „:
(I) Add vertices and edges.
(II) Let C, and C2 be two disjoint graphs. Let ax and />, be two adjacent
vertices in G1% and let a2 and ba Ik two adjacent vertices in Ca. Remove

CHROMATIC NUMBFR
3St
c<jgcs (fl|, hi] and [aa, /;,]. Add an edge [bl%ba], und contract set {a,, a3}
into a single vertex. Sec Fig. IS. 10.
(Ill) Contract a set of two non-adjacent vertices into a single vertex.
Clearly, if we perform these operations on graphs in '. ,, the resulting graph
belongs also to '. ,.
A,
rig. 15.10
I hcorcm 9 (Hajrts [1961 ]). Every graph G with >(C) > qivn be obtained from
(<l + I Viiiques by means of operations (I), A1) and A11).
Suppose that C ■ (A\ E) is a graph of order n which contradicts the
theorem, and assume that G has a maximal number of edges with respect to
this property.
I. We shall show that the relation "vertices .v and y are identical or non-
adjacent" is transitive. Suppose that this is not true: there exist three vertices
x, y. z such that (.v, y] g E, [y, z] ? E and [x, z] e E. Krom the maximality of
G, the graphs C + [.v, y] and G + [y. z] respectively contain graphs C, and
Gj that can be constructed from A'<+, by the operations. We may assume that
Gi and Ga arc disjoint (by duplicating). Let j.v,, j-J be the edge in C, that
corresponds to [.v, >•], and let [y2, z2] be the edge in Ca that corresponds to
[;'. r] (since [.v, y] appears in C, and [>',:] appears in Ga). Remove edge
l-vi.j'i] from C, and edge (j'a.rj from Ca, contract {yi.ya) into a single
vcrtex and join vertices at, and ia (i.e., perform operation (II) on graphs
ci and Cj). In the resulting graph G\ identify all pairs of vertices that cor-
fespond to the same vertex in C. which arc necessarily non-adjacent in
c' (operation (III)). The resulting graph G" is a partial subgraph of G.
"ence graph G can be constructed from graph G" by operation (I). This con-
•di the definition of G.
2- The above equivalence relation divides A" into classes St,Sa S*
Such that any two vertices belonging to distinct classes are adjacent and

3S2 GRAPHS
such that any two vertices belonging to the same class are non-adjacent
Since C cannot be constructed from /C,M by operation (I), we have C *
Kqtl. Thus k < q and, consequently.
which contradicts y(G) > q.
QE.D.
Using the Maj6s theorem, the four colour conjecture can be restated as
follows: Every graph obtained from 5'Ctlques by operations (/). (//) ami {III) fa
nonplanar.
Clearly, this is true if the construction is limited to operations (I) and (II).
Using the Haj6s theorem, the Hadwigcr conjecture can be restated as
follows: Every connected graph obtained from (q + I )-cliques by the thret
operations can be transformed into K,»+i by elementary contractions.
Heuchenne [1968] has presented more restrictive operations to construct
from KQtl a family of graphs G with >(C) > q that can be transformed
into K,., by elementary contractions. He conjectures that every graph C
with v(G) > q belongs to this family.
4. Chromatic polynomials
Let C - (X. E) be a graph with vertices xt. x3,.... .vn. In this section we
shall enumerate the distinct /.-colourings possible for graph G, i.e. the map-
mappings f(x) from X into {I, 2 X} such that
N [x.y]*E ~ f(x)+f{y).
77ie chromatic polynomial of C is defined to be a function P{G: X) that ex-
expresses for each integer X the number of distinct J.-colouring possible for
graph G. This number was originally expressed by G. BirkholT [1912] with
determinants. An excellent treatment of chromatic polynomials, due to
Read A968), serves as the basis for this presentation.
Fl8-lS.ll
Example I. Colour the graph G in Fig. 15.11 with X colours; vertex b
can be coloured first with any of the X colours, then vertex a or c can be

CHROMATIC NUMBER 3S3
cO|ourcd with any of the rcmiiining (X — I) colours. Hence, P{G;X) —
tf ' '>'•
Exampi E 2. Let C be the /i-cliquc A',. Vertex .Vi can be coloured first with
ony one or the /. colours. Then vertex ,v2 can be coloured with any one or
the remaining X — I colours. Then vertex x3 can be coloured with any one
of the remaining X — 2 colours, etc. Hence,
P(G ; X) - X(X - I) (X - 2)... (X - n + I).
This polynomial is often denoted by [>!]„, where [/]» - 0 for X < n.
Property I. Let a and b be two non-adjacent vertices in graph G. Let G
tit the graph obtained from G by joining a and b by an edge, and let G be the
graph obtained from G by contracting a and b into a single vertex. Then,
P(G ; A) - P(G; k) + P(C: X) .
Recall the principle of contraction-connection mentioned in Section I:
Clearly. P{G;X) equals the number of x-colourings in G for which a and /;
have different colours plus the number of A-colourings of G for which a
and b have the same colour. The formula follows.
Q.E.D.
II' the connection and contraction operations performed on graph G ter-
terminate with complete graphs Hx, H2, //3 then
P{G\X)- £P(W,;a).
i
For example, the contractions and the connections on the graph G in Fig.
15.2 terminate with five distinct cliques: one S-cliquc, three 4-cliques and
one 3-cliquc. Hence
; X)
Corollary. If G Is a graph of order n. the chromatic polynomial P(G; X)
h a polynomial of degree n in 1. The coefficient ofX"is+ I, and the coefficient
o/A° is 0.
Clearly, P(G; X) is the sum of terms of the form [X)p, where p assumes the
valuc n exactly once. Q.E.D.
Properly 2. If graph G has p connected components lft, //a,...,//,. then
P{G ; X) - />(//, ; X) P(Ht; X)... P{Hf ; X).
Clearly, each connected component can be coloured separately, and the
formula follows. Q.E.D.

354 GRAPHS
Property 3. If graph G contains an articulation set A that is a k-cHque
q pieces //,. //a,.... //„ relative to A, then
P(G;k) - ([*]»)' '"PHI, ;X)...P(Hll;l).
Colour A in one of the [k]* possible ways. Then there urc ..''
ways to colour piece //,, and the formula follows.
QF.D.
Theorem 10. The coifficients of P(G; k) are alternately non-negaiire ami
non-positive integers.
This result is a direct consequence of the Mobius function theory developed
by G. C. Rota [1964]. We shall give here an inductive proof on the order
n of G.
If n m I, 2, the theorem is obvious. If the theorem is valid for all graphs
of order < n. we shall show that it is also valid for a graph G of order iu
Clearly, this is trivial if G has only one edge. If the theorem is true for all
graphs of order n with less than in edges, we shall show th.it it is also true
for a graph G of order n with in edges,
If a and b arc two udjuccnt vertices in C. let G - G • [a, />], and denote
by G the graph obtained from 6* by contructing {a. h} into a single vertex.
From Property I.
P(G :/.) - P(G;X)- P{G;X).
Since the theorem is assumed to be valid for nil graphs of order n with
less than m edges, there exist integers a, > 0 such that
P{G;X)m)." -a,*"'1 +«,;"-' - •■•
Furthermore, since the theorem is assumed to be valid for all graphs of
order < n, there exist integers b, > 0 such that
P(G; k) - ;.""' - ft, k*-* + b2 ;.""J - -
Hence,
P(G \k)mkK- (a, + I) A" + (a, + bt) k"'2 - -
Thus, the coefficients arc alternately non-ncgativc and non-positive integers.
Q.E.D.
Corollary. If G is a graph of order n with in edges, then the coefficient of
kn~l in the polynomial P{G; k) equals — in.
Win m I, then P(G; k) - kn~l(k - 1). and the result follows. If m >

CHKOMAT1C NUMHFR .155
■t suffices to observe in the proof of theorem 10 that the addition of an
jgC increases the absolute value of the coefficient of A" by one unit.
Q.E.D.
Theorem 11. A graph G of order n is a tree if, and only if,
1\G ; /.) - >.(k - I)".
|. First we shall show that if G is a tree of order «. then the formula is
valid- Clearly, this is true for n - I and n - 2. If the formula is valid for all
trees of order < />, we shall show that it is also valid for a tree G of order />.
Since G has a pendant edge (Theorem 2, Ch. 3). Property 3 can be in-
invoked with the pendant edge as Hi and with the remaining tree of order
n - I as //a. Thus,
p(G;x) - <;.)'-J xa - i)xa - i)" - x(x - ir~l.
2, Let G be a graph of order n with P(G; X) - ;.(/. - I)". Graph G is
connected: Otherwise, it has p > 2 components //,, //a //„ and no con-
constant terms appear in the polynomials P(//,; /.) by the corollary to Property
I; by Property 2, P(G; X) equals their product and, therefore, could not have a
non-zero coefficient for X.
Since ;hc coefficient of A" is — (n - I), graph G has n - I edges (by
the corollary to Theorem 10), Hence, G is a tree,
Q.E.D.
The roots of chromatic polynomials have been notably studied by Bcrman
and Time [1969] and by Tuttc [1970].
S. Vertex colourings of planar graphs (abstract)
Recall from Example 2, Section I, that colouring the countries on a map
corresponds to colouring the vertices of a planar graph.
The four colour conjecture, i.e., a planar graph is 4-colourablc, has been
'lie subject of many works starting with an incorrect proof by Kcmpe in
1875. We shall not study here the recent developments of this extensive
tApic; however, let us mention the following results:
(I) A planar graph that does not contain four cycles of length 3 is i-colour-
(GrUnbaum [1963]).
This result is in some sense the best possible because there exist graphs
example. Kt) with chromatic number 4 that contain exactly four cycles
"Mcngth 3. This is an extension of the Grtttwch theorem: A planar graph
Without cycles of length 3 is 3-colourahle (Grbtxsch [1958]).

356
GRAPHS
B) The maximum value of the chromatic number of graphs that can
embedded in a orientahle surface of genus g > I equals
This result, known as the Hcawood conjecture, has been the object
of much mathematical research; a proof was recently discovered by Ringd
and Youngs [1969].
For the case of a plane or of a sphere (g - 0), this formula cannot be
used to prove the four colour conjecture. However,
Theorem 12. Every planar graph Is 5-cohurahle.
We shall assume that this is true for all planar graphs of order < n. and
we shall show that it is also true for a planar graph 0 of order n.
From Corollary 2 to Theorem B, Ch. 2), there exists a vertex x0 with
degree < S. Colour the subgraph Cx-('««) with the five colours a,, a2, a3,
a4, oB. The colour for vertex x0 can be chosen without difficulty unless
du(x0) - S and the S vertices ;,, y2. ya, yt. y6 adjacent to .v0 (in clockwise
order) all have different colours. Suppose that ;', has colour a, for i -
1,2 S. In the bicolour graph G(ilta3), the connected component that
contains yt also contains ;'a. (Otherwise, colours at and s3 could be inter-
changed in this component, and x0 could be coloured a,.)
Fig. 15.12
Similarly, the component G(xuat) that contains >>, also contains jv ^
Fig. 15.12.
Hence, vertices y% and y6 cannot be connected by a chain in the bicoloiif

CHROMATIC NUMBER 357
craph G(»a, aB); if the colours »3 and a8 arc interchanged in the component
of this gruph that contains ya, then x0 can be coloured aa.
Therefore, C is 5-colourablc.
Q.fc.D.
Corollary. //"C is a planar graph of order n, then
Let E,, S3, S3, St. S8) be a 5-colouring of G. Then,
n - £ | S, | < 5 «(C),
and the corollary follows.
Q.E.D.
Theorem 13. If G is a simple planar graph whose faces are triangles, and
if all the degrees arc multiple of 2 (respectively, multiple o/3). then y(G) < 4.
This result is a reformulation of Corollary I (respectively. Corollary 2)
of Theorem (8. Ch. 12).
Tlicrc is an enormous literature on the four colour problem, The reader
is referred to Ringcl [1959] and to Ore [1967].
EXERCISES
1' Use (he Brooks theorem to show thai a regular graph G wiih
y(G) + >{C) - « + 1 ,
» either
A) n isolated vertices, or
B) an fi-clique, or
0) an elementary cycle or length S.
2- A graph C with n vertices is said to be of type T3{n. p. q). where n > pq, it there
wist two partitions (d, CB C») and (D,. DB..... D,) of the vertices such thnt
A) max I D, | - I/,
B) max | C, | - p.
C) I C, ad, | < I for all/.y.
D) two vertices belonging to the same C, are adjacent.
E) two vertices belonging to the same D, are nun-adjacent,
Show that if C is of type 7,(«. p. q), then C is of type T3{». P. q) and that
- p -

358 CiKAPIIS
Show that for any two integers p and </ such that
p q < it + 1 .
there exists a graph G of type Tj(h,/>,</) with y(C') ■ p and y(C) - q.
(Finck II966D
3. Show that in a 4-colourable simple graph G. the edges can be coloured red and bin,
such that each triangle contains two blue edges and one red edge.
4. Show that in a simple planar graph the edges can be coloured red and blue such that
each triangle contains two blue edges and one red edge. (Schttuble |I968|)
5. Consider an infinite graph whose vertices are the pairs {p, i) of integers with p , q
and with an edge linking (/>, q) and (r, *) if q - r or if/> - s. Show that this graph con-
tains no triangles and that its chromatic number is infinite. (Frdos, Majnal |I96O|)
6. Consider an infinite family of sets {AJIel) where A, c {I. 2,...}, | A, | < cc. Form
a graph C whose vertices are the sets A, and with an edge joining every pair of vertices
whose corresponding sets meet. '<
Show that for each /, the complementary graph of the subgraph of G generated by
rM) has chromatic number < | A, |.
Show that an infinite graph C - (/, F) represents a family of finite subsets of A,2.
3....) if. and only if. for each / /,
/<Cro<«>) < « (Krewcra-s A9461)
7. Consider the graph of the queen's moves in a chess game: the vertices correspond to
the squares of an n x n chessboard, and two vertices are joined by an edge if a queen
placed on the square corresponding to the first vertex controls the square corresponding
to the second vertex. Call this graph CJ. Show that
y(G?) - 5,'
yich m n if it is not divisible by 2 or by 3.
y(CJ) ■ n or « + I if n + I is not divisible by 2 or by 3,
n < y(Gjp) < n + 3 in nil other cases.
It has been conjectured that y(C$) - n or n + t for ii > 3.
(M. R. Iyer. V. V. Menon |I966|)
8. If CJf is the graph of the king's moves on an « x it chessboard where n > 2, show
thnt y<C{) ■ 4.
9. If CS is the graph of the rook's moves on an n x n chessboard, show that
YlC!) m I, ,
10. If Gtf is the graph of the knight's moves on an n x n chessboard where « > '•
show that yCG?) - 2.
11. If d is the graph of the bishop's moves on an ii x ii chessboard, show that
y(C?) - n if ii is odd.
12. Show that if C is an elementary cycle with n vertices, then
P(C;A)-a— !)" + (-I)" U-I).

CHROMATIC NUMBER 339
13. Show thai if G is connected, (hen the absolute value of (he coefficient of A' in
(|)
(Rend [19681)
14. Show thai the smallest integer r such Hint Ap has a non-zero coefficient in P{C; A)
equals the number of connected components in C.
15. Use Theorem 4 to prove the following result: If p and q are two integers, and if C
jit ).graph without loops and with y(G) > pq, and if/is a real valued function defined on
,l,c vertex set A", then ilierc exists a path [a0, a,,.... a,] with
/(flo) < /(«>,) < - < /(«>,)
or there exists a path [iba, />,,.... 6,] with
/(W > /(*,) > - > /<M
(Chvital. Koml6s[l97l])
Show that if (*i, x3 ArP,.i) is a sequence of pq + I distinct integers, then this
sequence contains an increasing sequence with p members or this sequence coniains a
decreasing sequence with q members. (Erdos, Szekeres [1935])
16. Show that the simple graph C with y(C) ■ k and with the maximum number of
edges is unique. Show that this graph is the complementary graph of graph Cn,k defined
in Theorem S. Chapicr 13. (Tomc&cu)

CHAPTER 16
Perfect Graphs
1. Perfect graphs
For a simple graph G •* (X, E), let
a(G) denote the stability number.
0(G) denote the minimum number of cliques that partition X,
y(G) denote the chromatic number,
co(G) denote the maximum cardinality of a clique.
Clearly, a(G) < 0(G) since a stable set S can have at most one vertex in
each clique of the partition. Similarly, io(G) < y(G).
Graph G is defined to be a~perfect if
a(GA) - 0(GA) (A<zX).
Graph G is defined to be y-perfect if
yiGA) - u*GA) (A<zX).
Example I. If G is a bipartite graph, then we know that
<x(G) - 0(G).
from Corollary 2 to the Konig theorem. Chapter 7. Thus, a bipartite graph is
a-pcrfect. A bipartite graph G is also '/-perfect because if it has an edge, then
y(C) - 2 - w(C)
(since from Theorem 4, Chapter 7, C contains no triangles), and if G has no
edges, then y(G) - I - w(G).
Example 2. If G consists of an odd cycle of length Ik + I > 3 without
chords, then G is not a-pcrfect because »(G) - k and 0(G) - k + I- (A
minimum partition of G consists of k 2-cliques and one I-clique.)
Moreover. G is not y-pcrfect because y(G) - 3 and w(G) - 2.
The concepts of a-pcrfect and y-pcrfect graphs were introduced (Berg*
[1961J. 1962)) to conjecture that a graph is tx-per/in if. and only if '"' "
y-perfect.
360

PERFECT GRAPHS 361
An analytic formulation of this conjecture is due to Fulkerson [1971].
In 1970, Olaru and Sachs presented an interesting approach and finally,
Lovfisz proved the conjecture. (A proof is given in Chapter 20.) An easy result
is the following:
Theorem 1. A simple graph G is a-perfect if, and only if, Its complementary
graph G is '/-perfect.
Clearly.
0(GA) - y(GA).
Thus, a{GA) - 0(GA) is equivalent to io(GA) - y(CA).
Q.E.D.
Corollary. If either graph G or its complementary graph G contain an odd
elementary cycle of length > 3 without chords, then G Is neither a-perfect nor
y-perfect.
Let A be the vertex set of such a cycle of G. Then, from Example 2.
s(C) 0(GA\ (o(GA) y{GA). Thus, G is neither a-pcrfect nor y-pcrfect.
If the complementary graph G contains such a cycle, then it is neither a-
perfect nor y-pcrfect, and, from Theorem I, G is neither a-perfect nor y-
perfect.
Q.E.D.
Hi is result, and the study of various classes of perfect graphs (Bcrge
11%3]. [1967], [1969]) suggest the following conjecture:
The strong perfect graph conjecture. For a graph G. the following conditions
ore equivalent:
1I) G is x-perfect,
B) G is y-perfect,
C) G contains no set A such that GA or GA is an odd elementary cycle of
length > 3 without chords.
We have shown ubovc that (I) » C). B) => C). If C) - (I), then C) » B).
lf C) =» B). then C) (I). This conjecture is still unproved.
1 hcorcm 2. // G is a connected graph with an articulation set A that is a
clique, and if each piece relative to A is a y-perfect graph, then G is y-perfect.
Let G be a graph that satisfies the hypothesis of the theorem. It suffices to
show that
- y{G).

362 ORAPHS
Koj(G) - k, then there exists a ^-clique in at least one piece C relative
A, and y(C') - <t)(G') - k. Each other piece G" relative to A satisfies °
y(C) - w(G") < *.
Thus, G is ^-colourable, and
* - w(G) < y(G) < k .
Hence, aj(C) - y(C).
Q.E.D.
Theorem 3. If G is a conntcted graph with an articulation set A that is a
clique, and if each piece relative to A is an a-perfect graph, then G is a-perfect.
Let G be a graph that satisfies the hypothesis or the theorem. It suffices to
show that
»(G) - 0(G).
Let C\, Ca C, denote the connected components of subgraph Gx_ ,
and let
A,-{alas A, »(Cf(U ,.,) - or(GCl) } .
Two cases must be considered:
Case I. U?»i At A. Then, there is a vertex ae A with
»(Gc(U(.))-«(Gc,)+ I 0- 1.2 p).
Therefore, a maximum stable set 5, in GC|uu»> satisfies
I S, | - a(GC() + 1 ,
{ a } c S, c C, u { a } .
Set So - Uf-i 5, is stable in G. and
I So | - f <x(GCl) + I .
el
Moreover, since a partition of G into cliques can be formed with A and
with the 0{GCl) cliques of a minimum partition of Q. for / - 1.2, .-.»/'• **
have
0(G) < | tf | - £ fl(GCr) + 1-1 a(GCl) + 1 -
- I So I < »(G) < fl(G).

PERFECT GRAPHS
363
Hence, «(O - 0{G).
Case 2. Uf-i A, - A . Then, for all /,
Otherwise, there would exist a stable set S, c A, u C, with
I S, | - <x(GC() + 1 ,
and there would exist a vertex ae A, that is non-adjacent to some maximum
stable set of C(, which contradicts the definition of A,.
Hence,
0(G)
<x(G) < 0(G).
Hence. a(C) - 9{G).
Q.E.D.
The next three sections treat particular classes of a-pcrfect and y-pcrfect
graphs.
2. Comparability graphs
A simple graph G ■ (A', E) is called a comparability graph if it is possible to
direct its edges so that the resulting I-graph (A'. V) satisfies:
{x,y)eU, {y. z)e V —■ (x,z)e(J (transitivity)
(x.y)e(J — {y,x)iU (anti-symmetry).
Clearly, a bipartite graph is a comparability graph. Furthermore, each
subgraph of a comparability graph is a comparability graph.
The graph in Fig. 16.1 is not a
comparability graph because edges ah,
he, ai. he, ce can be directed appropria-
appropriately, but then it is impossible to direct
edge ef correctly.
Fig. 16.1

364 ORAPHS
Theorem 4. Every comparability graph is a-perfect.
Consider a l-graph G - (A', V) whose arcs represent an order relation
From Corollary 2, to Theorem A4, Ch. 13), »(G) equals the smallest number
or elementary paths that partition the vertex set. Because of transitivity, each
path generates a clique of G, and, by the corollary to Theorem D, Ch. 10) each
clique is the vertex set of some elementary puth of G. Hence, a(G) m ^v
Q-E.D. '
Theorem 5. Each comparability graph is y-perfeci.
It suffices to show that if G - (X, V) is the l-graph of an order relation,
then y(C) - w(C).
Let t(x) denote the length of the longest path from x plus one. Since C has
no circuits, t(x) < oo. If max t(x) - k, then there exists a fc-clique. There
exist no {k + l)-cliques (because this clique would contain a path passing
through all its vertices, and the longest path contains only k vertices). Thus
w(C) - k.
Consider k colours denoted by 1,2 k. and colour each vertex x with
colour t(x). Two adjacent vertices cannot have the same colour, because if
there is an arc directed from x to >'. then t(x) > t(y). Thus
V(C) < k
Since y(G) > <o(G) - k, we have
y(C) - k - u>(G) .
Q.E.D.
Theorem 6. Let G - (X, U) be a transitire l-graph, ami let E denote the
set of all pairs of adjacent vertices. Thai, the simple graph {X, E) is a compar-
comparability graph.
We shall remove from G all the loops and one arc in each multiple edge, so
that the resulting l-graph {X, V) satisfies the transitivity property.
Let *,. x2 .vB Ik the vertices of G, and let .v, ■ x, if either i ■ j or
/ ./. (xt,x,)eU, {x,.Xl) U.
From the transitivity of C, the relation ■ is an equivalence. Define set Vv&
follows:
A) lf.v,and*, are in the same equivalence class, and if/ > j.\c\.(xt,x,) v
B) If x, and x, arc in different equivalence classes, and if (.v,. x,) e V, 'et
(*,, x,)e V.

PERFECT GRAPHS 365
Hence we have:
(x, y) e V, (y, z)eV ■»■ {x, z) e V (transitivity)
(x, y)eV -» {y, x) i V (anti-symmetry).
Q.E.D.
Theorem 7 (Ghouila-Houri [1962]). A relation > is said to be a semi-order
relation if we have
a > b, b>- c ■* a > c or c> a {semi-transitivity)
a> b ■* not b >■ a (anti-symmetry)
A simple graph can be directed so that its arcs represent a semi-order relation
jf, and only if, it is a comparability graph.
It suffices to show that if G ■ (A\ V) is a l-graph whose arcs represent a
semi-order relation, then there exists an orientation of the edges of G that
represent an order relation.
Assume that the theorem is true for nil graphs of order < n; we shall
show that it is valid Tor a l-graph G ■ (A', V) of order n whose arcs represent
a semi-order.
First, note that if three distinct vertices a. h, and c satisfy
{a,b)eU, (b,c)eU, {c,a)eU,
then each other vertex is adjacent with either 0 or 2, or 3 of these vertices.
Suppose that G is not transitive; then there exist three vertices xx, xa, x3
satisfying
(*,.*2)e I/, (Xi.xJeU, {x3,xt)eU.
Two cases must be considered:
Case I. Each vertex x xt, xa, xa adjacent with one of these three vertices
'■' adjacent with all three.
Remove vertices x3 and x3 and direct the edges of the resulting subgraph so
that it is transitive. (This is possible from the induction hypothesis.)
Mr. 16.2

366 GRAPHS
Then, direct each edge of the form [.v, .va) or [x, xa], where x *,, *Si x
in the same direction as [.v, jti). and direct transitively the triangle formed bv
Xi, xa, x3. Thus, the obtained graph is transitive.
Case 2. There exists at least one vertex a Xi,xa,x3 that is adjacent h-///,
exactly two of these vertices, say. xa and x3.
Let A be the set of vertices y that satisfy
{y,x2)eU and {xity)eU.
Thus, *i e A. Furthermore, since a is not joined to xlt then a A. We
shall show that ifx A, then either x is adjacent with all the vertices of A, or *
is not adjacent with set A.
FIr. 16.3
Let be A. and suppose thut x is joined to a but not to b. By considering the
triangle formed by a. xa,xa, we see that x is joined to at least two of these
three vertices and thus to x3 or x3. By considering the triangle formed by b,
xa.xa, we see that .v is necessarily joined to .va and .va.
Since x is not adjacent with b. it follows that
{x,x2)eU and (xs,x)eU.
This contradicts xi A.
Now. direct transitively the subgraph generated by A and the subgraph
obtained by removing from G the vertices of A other than .v, (this is possible
from the induction hypothesis), then direct each edge of the form [x, ay
with xi A and ae A, in the same direction as [x. Xi). The resulting graph ■*
transitive.
Q.E.D.
A conjecture due to A. Hoffman can now be proved:
Theorem 8 (Gilmorc. Hoffman [1964); Cihouila-Houri. [1962)). A neces-
necessary and sufficient condition for a simple graph G - (X, E) to he a comparahilit"

PfrHFRCT GRAPHS 367
graph, is that for each pseudo-cycle [ax. fla °a«» «a«n • a\) of odd length,
there exists an edge of the form [fl,.fli+a) (where the addition Is modulo
A pseudo-cycle is a sequence of vertices starling and ending with the same
vertex such that any two consecutive vertices arc adjacent. For example,
graph G in l-'ig. 16.1 contains a pseudo-cycle
[a, b, c, d, c. e,f. e. b, a]
of length 9. but no edges of the form indicated. Thus, this graph is not a
comparability graph,
1. Necessity. If the graph (X, V) of an order relation contains an odd
pseudo-cycle [fli.aa «a<,»i.«i) without any edge of the form [a,,at + 3]%
then when this pseudo-cycle is traversed, arcs arc alternately encountered
along and against their direction. But, this is impossible if the pseudo-cycle is
odd.
2. Sufficiency. Consider a simple graph G — {X, £), and associate with G
the simple graph // - ( Y. F) dclincd by
| >e Y *» y~(a,b). a.beX, \a.b\*E
[) \[y./]eF o {)'./)-{{a.b),(b.c)}, [a.c)*E.
It is easy to show that the hypothesis implies that // contains no odd cycles.
and therefore, by Theorem D. Ch. 7). // is bipartite. Thus, the vertices of H
can be partitioned into two classes Yo and Y — Yo, with
y,y'eY-Y0 - [y./WF.
Direct each edge [a. b] of G from a to b (and write a > h) if {a. b)e Yo:
I'Vom (I), note that in graph // the vertices y - (a.b) and y' - (b,a)
'ire adjacent (because [a.a]iE. i.e. G has no loops); thus, these vertices
belong to two distinct classes, and each edge of G has received exactly one
direction.
• 'urthcrmorc. we have
or
(b,c)eY0

368 GRAPHS
Therefore, the relation > is a semi-order relation, and by Theorem 7, q j,
comparability graph,
QE.D.
Other characterizations of comparability graphs have been discovered b
Gallai [1967]. y
3. Triangulated graphs
A graph G is defined to be a triangulated graph if each cycle of length > 3
possesses a chord (i.e. an edge joining two non-consccutivc vertices of the
cycle). The concept of triangulated graphs is due to Hajnal and Surdnyj
[1958], A subgraph of a triangulated graph is also a triangulated graph
because, otherwise, it would have a cycle of length > 3 without chords,,and
G would also have a cycle of length > 3 without chords [
Example I. A tree is a triangulated graph,
Example 2. A cactus with only cycles of length 3 is triangulated because it
contains no cycles of length > 3.
Example 3. Graph L(G) is defined to be the graph in which each vertex
<F, represents an edge e, of G and with two vertices joined together if, and
only if. they represent adjacent edges in G. If G is a cactus with only cycles
of length 3, then the graph L(G) is triangulated: Otherwise, L(G) contains a
cycle [?i.?a **. ?i] without chords and with k > 3, and this cycle
corresponds in G to a cycle (<',. fa. e3 ek, e,) of length > 3. which con-
contradicts the definition of G.
The structure of triangulated graphs can be clarified by the following
theorem, due essentially to Hajnal and Surdnyi [1958].
Theorem 9. // G is a triangulated graph, then each minimal articulation set
of G is a clique.
Let A be a minimal articulation set of G. Removing A creates several con-
connected components C. C, C Each vertex a e A is joined to each of these
components. (Otherwise. A — { a) would be an articulation set of C, which
contradicts the minimality of A.)
Let o, and au be two vertices in A. There exists a chain n - [oi. cit ca, • •
cf, fla]. where

PKHFECT t.RAPHS 369
s that n is a chain of this type with minimum length. There also exists a
chain n' - ba, c\. c'a ri. a,], where
that ;<' is a chain of this type with minimum length.
The cycle ft + /i' has no chords of the following types:
- [fli.c,] (f* 1)
- [c<> cj] (' ** J) because // would not be the shortest chain
- [flj.cj (i p) .
_ [rji c'/] because C and C arc two distinct connected
components of GX.A
- [<"!• c'j] (' 7) ? because /i' would not be the shortest chain.
- ["i. OJ ^7 ?* a) '
Since the graph is triangulated, cycle /< + n', which has a length of at least
4, possesses a chord. This chord must necessarily be [«i. au].
Thus, any two vertices of A arc adjacent, and A is a clique.
Q.E.D.
Corollary I (Bcrge [I960]). A triangulated graph is y-perfect.
Clearly, this is true for graphs with I. 2, or 3 vertices. If this is true for all
graphs with < n vertices, we shall show that it is true for a graph G with n
vertices.
Suppose that G is neither disconnected nor a clique (because then the proof
would follow immediately). From Theorem 9, G contains an articulation set
A that is a clique. Each piece relative to A is y-pcrfect by the induction
hypothesis. Thus, from Theorem 2. G is y-pcrfect.
Q.E.D.
Corollary 2 (Hajnal. Surdnyi [1958]). A triangulated graph is a-perfvet.
The proof is identical to the proof for Corollary I, with Theorem 3 replac-
ln8 Theorem 2.
4- /-Triangulated graphs
A graph G is defined to be i-triangulatcd if each odd cycle n of length > 3
tas a set of chords E(n) that form with (i a planar graph whose unbounded

370 ORAFIU
face is the exterior of fi, and whose bounded faces arc all triangles. Th"
concept is due to Gallai [1962].
The following theorem gives several characterizations of /-triangulated
graphs.
Theorem 10. For a graph G, the following properties are equivalent;
A) C is l-triangulated,
B) For each elementary cycle n of odd length in G, and for each edge e of
ft. there exists a rertex of n that forms a triangle with edge e.
C) Each elementary cycle o) G of odd length k has k — 3 chords that do
not cross one another,
D) Each elementary cycle ofG of odd length greater than 3 has at least tv0
chords that do not cross one another,
E) If two distinct vertices x and y on an elementary cycle n >f odd Ityigth
are non-adjacent, then u contains two other vertices z and i (distinct
from x and y) that are adjacent and such that the four vertices x, r. y, t
are encountered on n in this order,
F) For each elementary cycle
U ■ [.Vo, Xi, A*j, .... .Vjj, Xq]
of odd length and for each index i such that [*,_,, x, + I] $S £. then
exists an index j ? I — I, i, i + I. such that (a1,. xt] - E.
The proofs for (I) B), B) D), C) D), (I) E). E) -F) are
immediate.
F) D) Let n - (.v0, xltxx, .... x^, x0] be a cycle in G of odd length 3.
If there is an edge [.v,, x,] E with j / - I. / + I. / - 2. i i- 2.
then one of two cycles determined by this edge and by u is of odd
length > 3 and has a chord; since this chord docs not cross chord
[.v,. x,]. condition D) follows. On the other hand, if each chord is
of the form [a1,, xti.3], condition D) also follows.
A) ■*■ C) Let fi be a cycle of odd length «. Let m be the number of chords of
E(n), Let / denote the number of triangles in this triangulation
of /j. From Euler's formula, we have n — (n + m) + (/ + 1) - *
On the other hand, we have 3/+ n - 2(m + n). Hence, '«
/i - 3.
D) *• (I) Let u be a cycle of odd length, and let £„ denote a maximum set
of chords that do not cross one another. The edges of n and of «

rrw'ECT or aprs 371
form a planar graph. Each bounded face of this planar graph is
cither a triangle or an even face. Since fi is odd, at least one of the
faces is a triangle. If not all of the hounded faces are triangles, then
an even face can be found adjacent to a triangle. By removing the
edge that separates them, an odd face having at least five sides is
obtained. Since this face has at least two chords that do not cross
one another, this contradicts that £„ is maximum.
Q.E.D.
Gallai [1962] proved: Every i-triangulatedgraph is x-perfect. The original
proof vvus very involved, but later, 1.. Suranyi [1968] presented a simpler
proof. In fact, Gallai proved a stronger result: If C is a connected i-triangu-
fated graph, at least one of the following three conditions is satisfied:
(I) G contain* an articulation set that is a clique,
(II) There exists a set A c X such that GA is bipartite and Gx-a Is com-
complete, and such that
a A. beX- A -*> [a, b]eE,
(III) lite relation "x — y or [x, y] E" is an equivalence relation.
Other classes of perfect graphs have been investigated, mid the following
results have been proved:
A) If each cycle of odd length > 5 has two chords that cross one another,
ihen the graph is ^-perfect (Olaru; in Sachs [1970]).
B) If each (Mid cycle contains an edge such that each maximal clique that
contuim this edge contains three vertices of the cycle, then the graph is v-perfect
(Bcrgc. Las Vcrgnas [1970]). The proof for this result in a different form is
Presented in Chapter 20.
5. Interval graphs
Consider a family .«/ ■ (/J,. Aa AH) of intervals on a line. The repre-
*entative graph of V is defined to be a simple graph G in which each vertex
"i corresponds to an interval A,, and with two vertices joined together if, and
On'y if, the two corresponding intervals intersect. Such a graph is also called
an interval graph. G. Haj6s 1957 and N. Wiener were the first to study interval
graphs, and so far two topological characterizations have been found; the
first is due to Lckkcrkcrker and Boland [1962], and the other is due to Gil-
j'torc and Hoffman [1964]. For bipartite interval graphs, sec also Koizig
11963],

372 okapiis
Kxample I. Each student visits (he university library once a day, and at
the end of the day he submits (he list of the names of the students met in thP
library while he was there. The problem is to find the order in which thestu
dents entered the library. Construct a graph G in which each vertex represents
a student, with two vertices joined together if. and only if, the corresponding
students were present in the library at the same time. This graph is an interval
graph, because it represents the intervals of time during which the students
were present in the library. The theorems of this section will give all the
possible solutions.
Exampi.f 2. In genetics, tests can be performed to determine if two
chromosomes overlap one another, and the problem is to prove or disprove
that a set of chromosomes arc linked together in linear order. Construct the
graph G whose edges arc the pairs of overlapping chromosomes; if this graph
is not an interval graph, it follows that the chromosomes cannot be linked in
linear order. v
Example 3. Consider the following problem that occurs in psychology:
Given a finite number of points x{, .va .vn on a line and an infinite family
Q of intervals, two points .v, and x, arc said to be indistinguishable if there is an
interval in family Q that contains both of them. The indistinguishable
pairs determine a graph, and we may ask for the characteristic properties of
these graphs.
In fact, such a graph represents a family of intervals /,. /., /„, where
interval /, corresponds to point x, and is defined by
/i ■ [.v(, + oo] n U { u), I a), e t). <a, 3 xt}.
If .v, < .v,, point x, and point x, arc indistinguishable if. and only if, x, A.
which is equivalent to /, n /, 0.
Theorem 11. Every interval graph G is triangulated.
Suppose that there is a cycle («,. a ap, «,] without chords. Let A, be
the interval corresponding to vertex «„ since interval Ak docs noi overlap
with interval Ak.t, the initial endpoints of the A, constitute a monotone
sequence; and therefore. Av cannot overlap with Alt which contradicts tb»l
[ap.<i,] is an edge of G.
Q.F..D.
Remark. The converse is not true: Graph C in Fig. 16.4 is triangulated, bul
we shall show that G is not an interval graph.
Clearly, the intervals A,,Ait Aa arc pairwisc disjoint and may be placed HJ
this order on the line. But, then, interval 0j that intersects intervals A\ a""

373
H«. 16.4
A3 must also intersect interval Aa, which contradicts that vertices at and b3
arc non-adjacent.
Corollary. Eeery interval graph G is st-perfecl and y-perfecl.
Since G is triangulated, G is a-pcrfect (Corollary 2 to Theorem 9). and y-
perfect (Corollary I to Theorem 9).
Q.E.D.
Appi ication I (Gallai). lfsJ is ajinitt family of intervals on a line, and if
k fs the maximum possible number ofpairwlse disjoint intervals in s#, tlwn there
txist k points on the tine such that each interval contains at least one of these
points.
Let graph G represent these intervals. Each clique in G corresponds to a
family of intervals having one point in common, by the I Icily theorem/1'
The result follows.
Application 2. ifsJ is a family of intervals m a line, and ifk is the maxi-
maximum number of intervals that together have a non-empty intersection, then the
intervals can be coloured with k colours such that no two intervals with the same
colour intersect.
The first application can be related to Example I. The minimum number
°f photographs of the library that arc needed so that each student is photo-
graphed at least once equals the maximum number of students who were
Nrwise not present together in the library.
We shall now study characterizations of interval graphs.
Lemma I. If G is an interval graph, then its complementary graph G is a
r"'nparability graph.
'" liclly'* theorem for intervals can be scaled as follows: If a family of inicrvnls
°°«s noi contain lwo disjoini iniervnl*. I hen nil the intervals hnvc n common poini, A
""iple proof Is given in Chnpier 17. Scciion 3. Example 1.

374 graphs
If graph C represents a family of intervals . two vertices x and y ar
linked together in G if. and only if, they represent disjoint intervals of
Direct edge [x, y] from x to y if interval y is to the right of intcrviil x on the
line. This produces a l-graph (X, U) such that:
(x. y)eU » {y,x)tU.
{x,y)eU.(y,z)e U m. (x.z)eU.
Thus, £ is a comparability graph.
QE.D.
The following lemmas treat u simple graph G with the following properties:
1I) £very cycle of length 4 has a chord,
B) the elementary graph is a comparability graph.
Let G - (X. E) be a graph with these properties. Let % denote the family
of the maximal cliques of G. Let ■ (X, V) denote the complementary
graph of G, assuming that the edges of £ arc directed transitively.
Lemma 2. Let C\, Ca e : there exists an arc in G that joins together set d
and set Ca. furthermore, all the arcs in G that join Ct and Ca hove the samt
direction.
If Cl and Ca arc two distinct maximal cliques of G, then C contains two
non-adjacent vertices a C\ and c e Ca. (Otherwise, C\ u Ca would be a
clique.) For example, let (a, c) V.
Let bd be another edge of C with b d, </c Ca and u b.
Ifc — (I. then edge bd has the same direction as edge ac (from Ct to Ca)
because, otherwise. G would not be a comparability graph.
If c d, then either ad or he is an edge of G (since, otherwise, the cycle
[a. d. c, h, a] of C would contain no chords). Without loss of generality. Id
this edge be ad. Then, (a, d) e U. Hence,
{b.d)eU.
rims, in both cases, edge bd is directed from C\ to Ca, as is edge ac.
Q.E.D.
Edge of G
— ►— Arc of 5
H«. 16.5

PERFECT GRAPHS
375
l,emma 3. Let H he a l-graph whose vertices represent the cliques off and
an arc from Ct to C2 if there exist two vertices aeCi and ce C3 in C
{a, c) € U. Then, II is a complete, transitive l-graph.
|3y Lemma 2, // is complete and anti-symmetric. It remains to show that if
(f i. C1*) 's an arc °f " an£l ^a« Ca) is an arc of //. then (Ci. C3) is an arc of
//.
We shall suppose that (C,, G) is not an arc of // (and we shall show that
this leads to a contradiction). Then, (C3, C\) is an arc or //.
For example, suppose that:
k. 16.6
o.fteC,, c.deC2, e,feCy
(b,c)eU, (d,e)eU, (f,a)eU.
d f C b hi h li
In this case, ad is an edge of C because, otherwise, ihc cliques \a.d) und C3
would contradict Lemma 2. Thus, from Lemma 2. («. d) e U. Since 1/ is a
transitive relation, (a. e) € U. But. then. (/. a)e U and (a. p) U implies that
(/, r) U. This contradicts that both vertices e and/belong to C3.
Q.E.D.
Theorem 12. (Gilmorc. HoiTman A964]). A simple graph G is an interval
graph if, and only if, the following l\\v conditions hold:
A) every cycle of length 4 has a chord,
B) the complementary graph i is a comparability graph.
Necessity. Condition (I) is necessary because u graph ili;U represents u
mily of intervals is triangulated (Theorem II). Condition B) is necessary
fr°m Lemma I.
Sufficiency. Let C - (X, £') be a simple graph that satisfies conditions (I)
a"dB). Let' - { f,,Cj C,} be the family of the maximal cliques of C.
^s in Lemma 3. form the l-graph //. Graph // is complete, anii-symmciric

376 GRAPHS
and transitive. From Theorem E, Ch. 10), //contains a unique hamiltonia
path n. Suppose thut the cliques of are indexed so that n - [Cx. Ca q %
Consequently. (C(. C,) is an arc of // if, and only if. / <./.
Note that G represents the sets
/, -{//C,€ ,C,bx),
because two vertices x and y are joined together in G if, and only if,
/, n Iy + 0 .
To show that ix is an interval, it suffices to show that
p < r < q J
If x Cr, there exists a vertex y € CP such that xy is an edge of C(see Fig
16.7).
C, Edge of C
I'lg. 16.7
Clearly. ix.y)e U (since r > p) jnd 0». .v)e 1/ (since r < q): this gives the
required contradiction.
Q.E.D.
6. Cartesian product and cartesian sum of simple graphs
In Chapter 14. three operations were defined for 1-graphs: normal product.
Cartesian product, and Cartesian sum.
If G - (X. E) and /■/ - (Y, F) are two simple graphs, the Cartesian sum
of graphs C arid // is defined to be a simple graph G + H whose vortex set t
X x Y. and with two vertices xy and x'y' joined together if. and only if.
cither x - *'. [y. y'] - F, or [x, x'] € E. y - / .

perfect graphs 377
Let the Kronccker delta be defined on graph G as follows: <5c(jc. x') - I
jf x x', and 50(jc. .v') - 0. otherwise. The number of edges thai join
vertices xy and x'y' in graph G + H can be written as
'»a*u(xy> x' >'■) - 6G(x. x') m,,(y, y') + 6a{y. y') mc(x, x').
The cartesian product of graphs G and // is defined to be a simple graph
q x // whose vertex set is X x Y, and with two vertices xy and x'y' joined
together if, and only if (.v. x'] e E and [y. y'\ e F. The number of edges joining
these two vertices in graph G x If can be written as
»'ox «(■*)•• *' /) - »'c(x, x') m,,(y, y').
The normal product (or simply, product) of graphs G and // is defined to be
a simple graph G.ll whose vertex set is X x Y, and with two vertices xy and
//joined together if and only if, cither
*-*'. [v.y1] e F.
or
[x.x']eL\ r-/.
or
[x.x]eE, [>-./]€/•'.
The number of edges joining these two vertices can be written as
»'<;,»(*, y) -
- mc(x. x1) m,,(y. y') + SG(x, x') m,,(y. y') + m^x. x1) <%,(y.y').
Note that these operations are commutative. If the definitions arc extended
to more than two graphs, it can be shown that the operations are associative
and distributive (C. Picard [1970]).
These definitions were first introduced to study the chromatic number and
the stability number. The relationships between these operations and the main
fundamental numbers arc described in this section.
Proposition I. Let G and H be two graphs. Then
w(G + H) - max{ w(C). «{//)}.
Lei Ci. Ca C, be the maximal cliques of G. and let /),. D3 Dv be
•he maximal cliques of H\ then the maximal cliques of G + H are the sets
C| * {)'/} and the sets {xt} x D,. Hence.
|{ *,} x DJ, | C, x { y, ) |} - inax{ ta{G). w(H)} .
Q.E.D.

378
GRAPHS
Proposition 2 (Vizing A963). Abcrth A964)). leiG and // be two graphs. Then
/J)-max{y(G),y(//)}.
1. Letr - max{y(C), y(//)};colourGand //with/colours0. 1.2.r- |
Let g(x) — k if vertex x is coloured k. For each xye X x Y, let
g(xy) m g(x) + g(y) (mod. r).
Consequently, g{xy) defines a colouring of C + // because if xy and x'y' are
adjacent and have the same colour, then either x «■ x' and vertices y and/
arc adjacent in //. or y ■ / and vertices x and .v' arc adjacent in G. In the
first case,
Six) + g(y) ■ g(x) + g{y) (mod.r)
i
and /?(.)') - g(y'), which is impossible because [y, y'\ e F. A similar result
follows for the second case. Hence. G + II is r-colourablc.
2. Graph G + // cannot be (;• - l)-colourublc. because, graph G H
contains a subgraph isomorphic to G and a subgraph isomorphic to H.
Hence. y(C + //) - r.
Q.E.D.
X/
a
MX. 16.8
a + //
Proposition 3. / r/ G «;«/ // he two graphs. Then
a(G + H)

PERFECT GRAPHS 379
]f 5 is a maximum stable set of C. and if F is a maximum stable set of //,
then the set S x T is a maximum stable set of G x //. Hence,
«(C + H) > | S x T! - «(C) «(//)
Q.E.D.
In Fig. 16.8, the vertices of a maximum stable set arc circled, and it is easily
seen that
<x(G + H) - 4 > <x(G) <x(H) m 3 .
Proposition 4. If graphs G and H r 'spectively hare orders n(i!)and «(//), then
«(C + H) < min{a(G)rt(//),a(//)fl(C)}.
Let So be a maximum stable set of G + II. Its intersection with set X x
{_>',} cannot have more than x(G) vertices; hence. \S0\ < | Y\a(G).
Similarly, | So \ < | X \ ot(//).
The formula follows.
Q.E.D.
Proposition 5. If graphs G and U respectively hare order ;r(C) and n{ II), then
0(G + II) < min{ n(GH(H), n{HH(G)} .
Let' -(Ci,Ca CJ be a minimum partition of G. The sets C x{^,}
arc cliques of C + // and cover X x Y. Thus,
fl(G + //) < q x | Y | - «(//) (KG).
Q.E.IX
Remark. Vizing [1965] proved a similar inequality for the dominance
numbers, i.e.:
(i*(G + H)
He also conjectured:
Conjecture. fi*(G + H) > /?*(C)/?•(//).
Ihcorcm 13. If only q colours are available, then the maximum number of
Vertices In G that can possibly he coloured with these q colours so that no two
adjacent vertices have the same colour is equal to x(G + Kv).
Let K, - (V. F) be a </-cliquc und let G - [X, E) be a simple graph. We
shall show that a stublc set of G + A', determines a set A <= V of vertices in G

380 ORAFHS
that can be coloured with q colours. (This result is illustrated in Fig. \f, g f
q - 3.)
Since the sets {xt} x Y arc all cliques in graph G + Kq, a stable set So of
G + Kv has at most one vertex in each of them. Put
A-{xfxeX; Son({*} x Y) + 0}.
For xe A, put g(x) - / if, and only if,
Son({x} x Y)m{Xyj).
Function g(x) is a ^-colouring of GA because two adjacent vertices x and
x in GA cannot have the same colour (since So is stable),
Conversely, \fg(x) is a ^-colouring of a subgraph GA. then the set of venices
xynx) with xe A is a stable set of G + Kv Thus, there is a one-to-one cor-
rc<pondcncc between the stable sets of G + K, and the partial ^-colouring
ofC l
Q.E.D.
Corollary. A graph G of order n isq-cohurable if. and only if, <x(G + Kn) - n.
The proof follows immediately.
For example, graph G in Fig. 16.8 is 3-colourablc, since
3t(G + KJ - 4 -
We shall now present a similar result for the cartesian product G x H:
Proposition 6. / el G and H be two graphs. Then
y{G x //)< min
Let q - min { y(G). y(//)}. and suppose that y(G) - q. Let g(x) be a
(/-colouring of graph G. Put
g(xy)-g(x) (xyeXx Y).
Thus. g(xy) is a ^-colouring of G x // because if xy and x'v' arc two adja-
adjacent vertices, then g(xy) g(x'y') (since .v and .v' are adjacent in G). Thus.
y(G x //) < q.
Q.E.D.
The following similar results arc available for the normal product G. H-
Proposition 7. Let G and H be two graphs. Then
G//)> max

PERFECT ORAFHS 381
Let q - max{ y(G), y(H)}. q - y(G)\ then q colours arc necessary to
colour G.H since G.H contains a subgraph isomorphic to G.
Q.E.D.
Proposition 8. Let G and H be two graphs. Then
cu(G\//)- ti){G)w(H).
Let C be a muximum clique of G, and let D be a maxiinum clique of H. The
set C x D is a clique of G.H because if x, x' eC and y. y' D. and if
xx' yy'- °nc of the three following cases occurs:
x - x' and [y, y'\ e F, and then, xy und jc'/ arc neighbours in G. H, or
y - / and [.v, *') 6 E, and then, xy and *'>•' are neighbours in G. //, or
[.v, .r1] e £ and [y, >■'] e /•". and then, xy and *'/ are neighbours in G. //.
Hence,
) > | C x D | -
Conversely, if Co is a maximum clique of G. //, let the projection of Co
on X be C, and let the projection of Co on K be D. Since C and /.) arc cliques
in G and H. the set C x /) is a clique in G, //. Hence. Co - C x I) (since
Co is a maximal clique). Thus,
co(C. //) - | C | - | C | x | D | < w(C) w(//).
The required equality follows.
Q.E.D.
Proposition 9. Lei G and H be two graphs. Then
If 5 and Tmc muximum stable sets respectively in G anil //. then
The cartesian product S x 7" is a stable set in G, //. and. consequently, we
have
/)> |Sx T\ -\S\ x in -3t(G)
Q.E.D.
An upper bound for x(G. II) will be given in Chapter 19. §2.
Proposition 10. 0(G. H) < 0(G) (HID .
Let (C,, Ca Cp) be a minimum partition into cliques of graph G Let
^i, Dq, ,,,, /)„) be a minimum partition into cliques of graph //

382
GRAPHS
In graph G.tf, set C, x D, is a clique, for / - 1,2 p, andy - 1.2,.. -
The cliques C, x D, partition graph G, //. Therefore.
0(G.H)*pq-0(GH(H).
Q.E.D.
Application (Shannon [1956]). Consider a transmitter that can emit five
signals, a. b, c, d, e, and a receiver that can interpret each of these signals in
two different ways. Signal a can be interpreted as either p or q, signal b can
be interpreted as cither q or r. etc.. as shown in Fig, 16.9. What is the maxi-
maximum number of signals that can be used for a code so that there is no possible
confusion on reception? This problem reduces to finding a maximum stable
set 5 of a graph G shown in Fig. 16.10, in which two vertices are adjacent if
and only if they represent two signuls that the receiver can confuse. For
example, we can take S - {a,c), since graph G in Fig, 16.10 has stability
number a(G) ■ 2, '
Mr. 16.9
Mil. 16.10
a.a "(il
e —'
\
s
X
A
(^—$
X
8
8
V
X
i )
X
x
8
X
k V
X
X
1
X
V
x,
X
X
,/
\ ™
\
#
abed e
Mr. 1A.11

PERFECT GRAPHS 383
Instead of single letter words, we could use a code of two letter words,
provided that no two letter words of this code can lead to confusion on
reception, Thus, the letters a and c which cannot be confused can form the
code; aa, ac. ca, ec which has a vocabulary of (a(G))a — 4 words. Hut an even
richer code is: aa, be, ce, db, ed. It is easily seen that no two of these words can
be confused by the receiver, This gives a vocabulary of 5 words.
Note that the words xy and x'y' can be confused if, and only if. these two
words are adjacent vertices in the normal product G. G - Ga, A code con-
consisting of 2-lcttcr words has a maximum vocabulary of s(G8) words. More
generally, the maximum possible vocabulary for a code of A- letter words is the
stability number of the product.
C* - G.G.G...C.
Jk
With this as motivation, the capacity of graph G (or." zcto-cttor capacity")
is defined to be the number
c(G) - sup v «(G*) •
The capacity of the graph G* in Fig, 16.10 is known to be between 2 and 3:
however, its exact capacity remains unknown. Furthermore, Ljubich [1964]
has shown that "V»(G)* tends to c(G) when k tends to infinity."*
An important problem is the characterisation of the graphs whose code is
improved when longer words arc used, We shall prove the following result,
due to Shannon:
// a graph G satisfies s(G) - 0(G), then the code cannot be improved by
taing longer words.
From Propositions 9 nnd 10, we have
< fl(G')
'" Fckcie's theorem slates ihm if n sequence of numbers a,, a3l... is sub-additive
(I.e., am»n *aK + a*), ihen
" ■* ?■•
If we lei an - -Iob o(C"), then a sub-additive sequence is formed because
«(C""»)>a(C)aCC«),
log a(C""") > log «<C"') + loga(C).
Thus,
log v *(C") ■* «up log V«(C")
and

384 GKAPHS
Hence,
Thus. «(C) ■ 0{G) implies that
"V
and the code cannot be improved.
Q-E.D.
In particular, when each signal is determined by its modulation frequency,
two signals can be confused if. and only if. the corresponding frequency
intervals of the two signals intersect ("linear noise"). In this case, graph C
represents a family of intervals, and therefore, by the corollary to Theorem 11,
a(C) — (KG)- Hence, if noise is linear. Own a code cannot be improved by using
longer words.
EXERCISFS
1. A graph G Is defined to be vtriex-crlilcal a-imprrfeel if it is not o-pcrfecl. bul if ih«
removal of any vertex makes the resulting gruph o-pcrfect.
Show that nuch a graph G satisfies the following properties:
A) for each vertex, there exists a maximum stable set that does not contain this
vertex,
B) for each vertex, there exists a minimum partition of C into cliques such thut one
or the cliques contains only this vertex.
E. Olaru has shown that each vertex x belongs to a cycle or length > 5 such thul all
the chords of the cycle arc incident to jr. (Sachs |I97O|)
2. Graph G is defined to be eitge-otitcul n-lmptij'ect if C is not «-pcrfcct but if the graph
G - « is a-perfect for each edge t of G.
Show thul such a graph is a-critical (see Chapter 13), Show I hut a connected graph
G is edge-critical n-lmperfcct, if, and only if, C is an odd cycle of length > 5 without
chords.
Him: Use Corollary 3 to Theorem F, Ch, 13).
3. Let Cr be a cycle of length 7, and let G - C-, be its complementary graph. Show that
A) G has no odd cycles of length > 3 without chords,
B) G is vertex-critical a-imperfect.
4. Consider a simple graph G - {X, E) with the following property: if x,, xt, Xi. £
arc four distinct vertices with [xt, xa] e £. [xa, xi] E, (,va, *«]« £, then [x,, x>) &
or (jfa, xt] fi E.
Show that graph G contains a vertex a thul is adjacent to every other vertex ol m*
graph,
Show (by induction on n) that the complementary graph or G is a comparability grapn-
(Wolk [19651)
5. Show that a simple graph G is the transitive closure of an urborcsccncc if, and only
if, it satisfies the two Following properties:

PERICCT GRAIW 385
A) C is connected,
B) Ifxi,xj,.v1,.t«arcroiirdi*linct vcrtices\vith(xi, .*.,] £, [*3.*j) F, [jr.,, jr,] c £,
then [jfi, x,\ «■ £ or [.v3, *,] e E,
////i/,' The sufficiency can be proved by induction on n. and by using Exercise 4.
(Wolk 11965))
(,. Show that iffl - u(C), ihen
7, If/>(G) denote* the number of connected component
/>{G '■ //)
t. Show thul if C and // are two simple graphs wiih hamiltonian cycles, then uruph
C * H has a hamiltonian cycle.
Show that if both G and H have culcrian cycles, Ihen graph G + H has an culcriun
cycle, (Abcrth [1964])

PART TWO
Hypergraphs

CHAPTER 17
Hypergraphs and Their Duals
I. Hypcrgraphs
Let X ■ {*,, xa xK } be a finite set, and let > ■ (£, / /e /) bs a family
of subsets of X. The family > is said to bs a hypergraph on X if
A) £( 0 U el)
B) U Ej - X .
The couple // - {X, ) is called a hypergraph. \ X | - n is called the ««/«■
of this hypergraph. The elements .v,. jra .vn arc culled the vertices and the
sets £,. £2 £„ arc called the edges.
A hypcrgraph is shown in Fig. 17,1, An edge £, with | £, | > 2 is drawn as
a curve encircling all the vertices of £,. An edge £, with | £, | ■ 2, is drawn
as a curve connecting its two vertices. An edge £, with | £, | ■ I. is drawn as
a loop as in a graph.
If the edges £, arc all distinct, the hypcrgraph is said to be simple, and <? is
a set of non-empty subsets of X. if | E, | -2 for all /, and if the hypcrgraph
'/ is simple, then // is a simple graph without isolated vertices. The main
purpose of the theory of hypergraphs is to generalize results from graph
theory.
In a hypcrgraph, two vertices are said to be adjacent if there is an edge £,
that contains both of these vertices. Two edges arc said to be adjacent if their
intersection is not empty.
The incidence matrix of hypcrgrnph H ■ (X. A) is a matrix {(a1,)) with m
r°ws that represent the edges of // and n columns that represent the vertices
°f'/. such that
-1 if XjeE,
-0 if XjiE,.
Each @. i)-matrix is the incidence matrix of a hypcrgraph if no row or
column contains only zeros.
To each hypcrgraph // - (X; Ex. £a £m) there corresponds a hypcr-
389

390 IIYPEHOKAPHS
graph //• - (£; X,, Xa AT,) whose vertices arc points ex,et,
(that respectively represent Ex. E% £„) nnd whose edges arc sets A',. \m
..,, Xn (that respectively represent ,v,, xa .vn). where, for all/ *'
X, - {e,ft X in. EtBXj).
Thus. A", ?* 0 and U' **; ■ & a»d //* is a hypcrgraph.
Hypcrgraph //• is called the dual hypergraph of II. The incidence matrix of
hypcrgraph //* is the transpose of the incidence matrix ((a/)) of hypcrgraph
//. and consequently, (//*)* - //, If two vertices x, and xK in // are adjacent,
then the corresponding edges X, and Xk in //♦ arc adjacent. If two edges £;
and E, in // arc adjacent, then the corresponding vertices e, and e, in //* are
adjacent.
In hypcrgraph //, the rank r(S) of a non-empty set S c A* is defined to be
the positive integer
r(S) - max | S n E, \.
i
The number r(X) is called the rank of hypergraph II.
If E, — r(X) for each /, then // is called a uniform hypergraph of rank r{X).
Thus, each simple graph without isolated vertices is a uniform hypcrgraph of
rank 2.
The partial hypergraph of II ■ (X, £) gincraled by a family . c is de-
defined to be the hypcrgraph (X. ., ). where X - Um £i- The subhyptr-
graph of II - (X, ) generated by the set A c X is defined to be the hypcr-
hypcrgraph IIA - {A, A), where
tA - { E, n A I E, e S ; £, n A + 0 }.
Proposition I. If II - (X, ) is a hypergraph with rank function r(S). the
subhypergraph IIA - {A, ffA) generated by a set A c X is a hypergraph with
rank function rA(S), where
rA{S) - r{S) (S c A).
The proof is immediate.
Given an integer k > 0, the k-section of hypcrgraph // - (X, «f) is defined
to be the couple HiM - {X, lh}) formed by X and the set
«*(*> -{F/FcX;l<|F|<fc;Fc£, for some E,et} •
Proposition 2. // // is a hypergraph with rank function r{S), the k-section
Hm of H is a simple hypergraph with rank function rM{S) where
rlk)(S) - min{*,r(S)}.

HYPEHOHAPHS AND THEIR DUALS 391
The proof is immediate.
Thus, the 2-scction //(8) of a hypcrgraph // is a graph with the same vertices
flS // and with a loop attached to each vertex. Let (//)., denote this 2-scction
without its loops. Clearly, (//)a is a simple graph.
2. Cycles in a hypcrgraph
In a hypcrgraph // - (X, ). a chain ofl'ngthq is defined to be a sequence
(x,, F,. .v,, £a £,. *,♦,) such that
A) xx, x2 xq are ail distinct vertices of II.
B) E,, Ea £, arc all distinct edges of H,
C) .vv. *ko e Ek for k - I. 2 q.
\{q > I and .va,, ■ .v,, then this chain is called a cycle of length q. If // is
a graph, these definitions correspond to those for an elementary chain and an
elementary cycle (except that a loop is not considered as a cycle in the theory
of hypcrgraphs).
Fig, 17.1. Connected hypcrgruph (AT: £,, £«, £„, £",, £tl £.) of order 8 wiih a
unique cycle ami wiih ^A /", | - I) - 2 + I + 2 + 2 + 1 + 0 - 8.
If there is a chain in the hypcrgraph that starts at vertex a and terminates at
vcrtcx h. then we shall write a m b.
Imposition 3. Hie relation a ■ h is an equivalence relation, whose classes
are called the "connected components" of the hypergraph. IJ C, Is a connected
v°ntptment that Inl rsects edge E, then C\ contains E.
It suffices to show that this relation is an equivalence
A) a m a, because there exists a chain of length I beginning and terminating
at vertex «,
B) a m b implies b m a.

392 HVPIRORAPHS
C) a m b, b ■ c implies a m c, because if there exists a chain of length »
from a to b and a chain of length q from b to c, then the edges of these
chains form a chain of length < p + q from a to c.
The theorem follows.
QE.D.
Proposition 4. If II is a hypergraph with n vertices, m edges and p connected
components, it contains no cycles (f, and only if,
I (I E, | - 1) - n - p.
i
Consider the bipartite graph G{ll) whose vertices represent the vertices and
edges of //, where the vertex representing xf is joined to the vertex represent-
representing Et if, nnd only if. .v, £,. Graph G(H) has p connected components,
mt + n vertices, and 2™-i I Et I edges. v
Hypergraph // contains no cycles if, and only if, G{fl) is a forest, i.e. ir
ID
(-1
or
£, I - Un + n) - p
I (|£,|- D-zi-p.
Q.E.D.
Proposition 5. // // - {X, «f) is a hypergraph with rf -(£,// e /), then H
has no cvcles if. and only if,
\\JE,\>l(\E,\-l) (Jc/:J 0).
\ i j | itJ
I. If H contains a cycle (o,. £,, a3. fa £,. o,), then, by letting Q -
{ 1. 2 (,-), we have
and the inequality of the proposition cannot hold.
2. If // - (X, ) contains no cycles, then a partial family (£, / i J)als0
contains no cycles. If this family forms a hypergraph with p connected com-
components, then by Proposition 4.
Q.E.D.

HYPERORAPHS AND THIIR DUALS 393
Proposition 6. A hypergraph (X, ) with p connected components and n
vertices has a unique cycle if, and only if,
I (|£,|- l)-«-p+ I.
1-1
The bipartite graph G(ll) defined in Proposition 4 has p connected com-
components, in -f n vertices, and 2, | £', | edges. It has exactly one cycle if, and
only if. its cyclomatic number is
Q. E.D.
theorem 1 (Lovasz[1965J). If a hypergraph II - (X. ) of order n with p
connected components contains no cycles of length > 3, and no loops, and if
\Etr\E,\K2 for E, E,,
(hen
f (I E, | - 2) < n - p.
i-i
1. If ;i — 2, then // has exactly one connected component, and, therefore,
the condition holds. Suppose that the theorem is true for all hypcrgraphs with
less than n vertices, n > 2. Assume that it is not true for a hypcrgraph H with
n vertices, and we shall show that this leads to a contradiction.
2. Hypcrgraph // is connected because, otherwise,
£ (| f,| - 2) < 11\\- 1.
for each connected component Ck. and by summing over k, the inequality is
obtained.
3. We shall show that //contains no vertex whose removal disconnects //
("nrticulation vertex"). If such a vertex .v0 existed, and if its removal created
two connected components C\ and Ca. then by letting D, - C, u {.v0}. we
*oulcl huve:
I (It, 1-2) < | Oil I.
K,~U,
I (| E, I - 2) < | D, | - I .
l,-Oi
Since // has no loops, there is no edge contained in both Dx and

394 HYFERORAFHS
D2. Furthermore, each edge intersecting D, (rcsp. Da) is entirely contained in
Dx (rcsp. Da). Thus.
fll£il-2)- I (|£,|-2)+ I (|£,|-2)<
1-1 EieDi E(cDj
4. We shall show that no vertex *, belongs to only one edge. If xx £, ^
Xi 4 £, Tor / »* 1. and if | £, | > 3. then the hypcrgraph
(£,-{*,}.£: £„)
contains no cycles of length > 3: then, by the induction hypothesis.
I (I £, I - 2) - l< (/1 - 1) - 1 .
which implies that the inequality of the theorem holds. If | £, | -2. the same
result is obtained by considering the hypcrgraph (£a. E3 £m).
5. We shall show that | £, n E, | - 0 or 2 for £, E,. Suppose that
£1 n £a - {x0}. Since | £, | > 2. consider a vertex a e £\ with <7 x0. and
a vertex /; € £3 with /; # x0. Since x0 is the only vertex of ^i n £3, it follows
that o b.
Since x0 is not an articulation vertex from Part 3. there exists a chain
(«.£•,. x;.£i x;. £;./>)
connecting a and b that does not use vertex .v0.
If there existed two indices /and,/:such that £,' - £, and £/ - £2. then the
cycle
(Xq, Ef, X|+1, £j+1, ..., Ej, Xq)
would be of length > 3 (because / + I ./).
If there existed an index / such thut £,' - £, and no index j such that
E', - £a. then the cycle
(.v0. £;.*,'+,.£,'+ £p. b, E2.X0)
would be of length > 3.
If there existed no index / such that £,' - £, or £,' - £a. then the cycle
(fl.£',.x', £;. b.£a.x0.£|.a)
would be of length > 3.

HYFERORAFH9 AND THEIR DUALS 395
In all cases, there is a cycle of length > 3 in H, which contradicts the hypo-
thesis.
6. Consider the family - { F\ F - £ n £', E. £' € . E n £' 0 }
^hcre each set contains exactly two vertices. We shall show that is a parti-
partition of X.
Clearly. Un Fm X because, from Part 4. each vertex x € X is covered by
ai least two edges £, and E, e <5\ and E,n E,6 .
Furthermore, let F and F' be two difTerciU sets of with
F - E.nf,, F' - £', n £a.
inhere exists a vertex x0 fn F', we can write:
F-{xo.x, }, F'"{x0. *i}. x, # x,1.
If £, - £i, hypcrgraph // contains a cycle (x0. £a. x,. Ex,x\.E'2, x0) of
length 3. which contradicts the hypothesis. If £t E[. £3, and if £3 »& £|, £8
let
£, n £', - { at0. fl| } £2 n £i - { x0. a2 }.
Clearly. ay at because, otherwise, Ex r\ E2 — E[ n £3. Therefore, we may
assume that, say, ax xj. Thus, there is a cycle (x0. £,, ax, E[, x\, E'2, x0)
or length 3. which contradicts the hypothesis.
7. Construct a bipartite graph G, whose vertices represent the m edges of
£ in one class and the sets of in the other class, and where the vertex repre-
representing £, e and the vertex representing /",€• arc joined if, and only if,
£. n F, 0. Note that, in this case, F, <=■ £, because, otherwise, F, -
E, n £; with E,, £J £,, and therefore £, n £, and Fy could not be two
different classes of .
Thus, the number of edges in graph G is
m(G) - 5 f I £, I.
From Part 6. the number of vertices in G is n(G) - '" * " •
8. Note that gniph G contains no elementary cycle [F^ £lt F3, £3
^k, FJ, because then k > 4. and if we select one x, in each F,, the cycle
(•vi. £1, xa, £;, £„, x,) would be of length > 3, which contradicts the
hypothesis.
I'rom Proposition 4, graph G satisfies
m(G) < h(C) - I.

396 HYPERORAPIIS
Hence.
2 (£ I E> I * '" + 2 ~ '
and
f (| £, | - 2) < n - 2 < n - I .
i-1
Q.E.D.
In particular. Theorem 1 shows that if // is a uniform hypcrgraph of rank 3
without cycles of length > 3, then
m < n - 1 .
3. Conformal hypergraphs
Consider a simple hypcrgruph // — {X, ) where ■ { £i. £a £„,}.
Consider the 2-scction (//)a. which is a simple graph. Each edge of (//), is
contained in at least one of the cliques £i, £a £„. Hypcrgraph Hh said
to be conformal if the set of all the maximal cliques of graph (//)a is equal to
the set > „,,„ of maximal edges of //. If // is a conformal hypcrgraph, each sub-
hypcrgraph of // is also conformal.
Umma. Hypergraph II is confornwl if, and only if, each clique of graph
(//)a is contained in an edge of II.
If // is conformal. it is evident that each clique of (//)a is contuincd in a
maximal clique of (//)a and thus in an edge of //.
To prove the converse, let m», be the family of maximal cliques of graph
(//)a. and let fim*x be the family of maximal edges of //. If Cer „,„. there
exists an edge E e <?„,„ and a clique C e with
C = E c C .
Hence. C - £. Thus. «•„„ c <fmM.
Conversely, if £e „,»„. there exists a clique C mM and an £'6 with
£ c C c £'.
Hence. £ - C. and Vm,x c mM. Therefore. „,„ - mM. and // is a con-
conformal hypcrgraph. '
Q.F.D.
Theorem 2 (Gilmorc [I960]). A necessary ami sufficient condition for a

HYPERGRAPHS AND THEIR DUALS 397
hypergraph H — (X, ) to be conformal is that for any three edges Et, £a. £3
cfH. there exists an edge of // that contains the set
(£, n £2) u (£2 n £,) u (£, n £,).
Necessity. If // is conformal. each vertex or £1 n £a is adjacent to each
vertex of £a n £3 and to each vertex of £3 n £,. Thus, set
(£, n £2) u (£, n £,) vj (£, n £,)
is a clique of graph (//K. From the lemma, this clique is contained in an edge
of//.
Sufficiency. We shall show that each clique C or (//)a is contained in an
edge of H.
If I CI < 2. this follows from the definition of the 2-section. Suppose that
[his is true for each clique C with \C'\ < k \ we shall show that it is also true
for a clique C with | C \ - k > 3.
Let Jfi. Jfa. *3 C and let C, - C - { x,}. By the induction hypothesis,
there exists an edge £, => Q. Thus, there exists an edge £0 of H such that
C-(Cxn CJ v (C, n C3) u (Ca n C3) c
c (£, n £,) u (£, n £.,) vj (£2 n £3) c £0 .
Q.E.D.
A hypcrgraph rf1 - (£, / /e /) is said to satisfy the lltlly property if 7 c /
and £, n E, 0 for all /, /£i implies that
n £j 0-
The following is the dual result to Theorem 2.
Theorem 3. A hypergraph 11 is conformal if. and only if its dual //* satisfies
ihc Hilly properly.
I. Suppose that H* satisfies the Helly property. We shall show that a
clique { *,, xa xk } of graph (//)a Is contained in an edge of H. Any two
vcrtices x, and x, in this clique arc both contained in some edge £„ of //.
Consider the edges X, - { e, fj < m, E,ex,} of the duul hypcrgraph //•.
Nearly. gp X, r\ X,. The sets Xit Xa Xk arc pairwise intersecting, and
'hcrcforc. by the Hclly property.
nx, 0.

398 IIYPHRORAPHS
If e, is a vertex in this intersection, then
F't 3 { xl • Xl Xk I >
and £, is the required edge.
2. If // is a conformal hypcrgraph, consider a family (X%, Xa xk) Of
edges in //* which are pairwise intersecting. This family corresponds to a
clique {xx, x2 xk) in (//K. and there exists an edge E, of // thiit contains
this clique. Hence,
Thus, //* satisfies the Hclly property.
QE.D.\
Corollary. A family {Xx, Xa>.... Xn)of subsets of a set E — {ex,e3 em)
satisfies the Hclly property if, ami only if, for each triple (e,, e,, <»*), the family
of the Xp that contain at least two of these three vertices has a non-empty inter-
intersection (if this family Is not empty).
Consider the hypergraph //■(£: Xi, Xa,.... Xn). From Theorem 3. the
Hclly property is satisfied if. and only if. the dual hypcrgraph //*
(X; £|. Ea Em) is conformed, therefore, from the Gilmore theorem, if for
each triple (<>,, et, ek) there exists an index s such that
E, => (£, n Ej) u (£'y nEK)^j(Ekn E,).
Let P be the set of indices p such that Xp contains at least two of the three
vertices e,, e,, ek. Thus, the above condition is equivalent to
x, e E, {pe P)
for some index .v, which is equivalent to
n x, 0.
Q.E.D.
Example I. Let £ - { Ci,ea en} be a finite set of points on a line. We
shall show that a family (Xlt Xa Xn) of intervals of points satisfies the
Hclly property.
Consider three points <»,. ea, c3, with c, < e9 < ea. If interval Xp contain

IIVPERCJRAPIU AND TIIFIR DUALS .199
pyo of these three points (write this asp e P), then it contains e3, and. there-
fore.
n xf 0-
Thus, from the above Corollary, the family satisfies the I Icily property.
Example 2. Consider the hypcrgraph( X; £,. £"a. .... Em) that is the dual of
aliypcrgraph (E; Xlt X2 Xn)defined as in Example I. We shall show that
this dual hypergraph also satisfies the Holly property. Consider three vertices
Xi,xi,xie X.
Let P denote the set of indices p such that Ev contains two of these three
points. We shall show that
fl£p 0 •
ft
We may assume that none of the intervals Xx. X2. X3 contains another,
because if X3 ^ A\. then each ep that belongs to two of the three intervals
belongs to X3, and
Among Xx, Xa, X3. let Xx be the interval whose initial endpoint is the
leftmost, and let Xs be the interval whose terminal endpoint is rightmost.
Consequently. Xa => A\ n ^3. and each e9 that belongs to two of the inter-
intervals also belong to X2. Hence.
Thus, the £, satisfy the Hclly property.
Example 3. Consider a tree G - (X. U), and let (Atllei) denote the
family of subsets A{ of X such that the subgraph GA, is also a tree. We shall
show that this family satisfies the Hclly property.
Consider three vertices a. b, c of G. Let P be the set of indices p
such that Ap contains two of these three vertices. If the three vertices belong
to the same elementary chain of G. then one of the vertices, say b, is between
'heother two in the chain. Thus, be flp.r'V
If the three vertices do not belong to the same elementary chain, then there
exists a vertex r of G that separates any two of them. Thus.
re PI A
r
,

400 HYPERGRAPHS
In both cases. f|,, fAr 0. Thus, from the Corollary to Theorem 3. the a
satisfy the Hclly property. '
4. Representative graph of a hypcrgraph
Let // - (£: Xx, Xa Xn) be a hypcrgraph with n edges. The represents
live graph of H is defined to be the simple graph L{H) of order n whose ver.
tices jci, x2 x* respectively represent the edges Xlt Xa,.... Xn of Wand
with vertices .v, and x, joined by an edge if, and only if, X, n X, 0.
For each simple graph G, there exists a hypergraph // such (hat G m £(#)
More precisely, we have the following:
Proposition 1. Let G be a simple graph with vertex set X,andlet(Ex, £a
£„,) be a family of subsets of X with the following properties:
A) each E, is a clique of graph G,
B) each vertex and each edge of G is covered by at least one E,.
Then, G is the representative graph of the dual II of the hypergraph
//•-(*;£,,£, £J.
Conversely, ifG is the representative graph of a hypergraph
H ™ (£; X|, X},..., Xn),
Own the dual H* -(#:£,. £a £n) satisfies properties (I) and B).
G - L(H) means that vertices .v, and x, arc adjacent if. and only if. there
exists in // a vertex e*e X, n X,. This is ulso cquivulcnt to saying that there
exists an edge £„ in //* that contains both .v, and x,, or to saying that
G - (//*J.
Clearly, this can be expressed as //* satisfies (I) and B).
Q.E.D.
For example, consider the graph G shown in Fig. 17.2. Graph G is the
representative graph for each of (he following hypergraphs:
A) (he mulligraph //, ((he £, arc the maximal cliques of G),
B) the hypcrgraph H2 (the E, are the edges of G),
C) the simple bipartite graph H3, etc.
Let ii(G) denote tlie minimum order of (he hypergraphs // for which
G - L(ll). For graph G in Fig. 17.2. we have fl(G) - 2 because G - WU)-
The following proposition shows that the determination of ii(G) reduces
to the determination of a chromatic number.

IIYPERORAPIIS AND TIIDR DUALS
G
401
Aihc C
Flu. 17.2
HX»(A,B,C.D.F.)
Order - 2
de II. - [A, B. C. D. E)
Order - 6
> abc
Order - 6
FIs. 17.3
Hropositioii 2. Let G be a simple graph with vertices JC|, xa xn, none of
*'hich is Isolated. / etGbea graph, each of whose vertices represents an edge of
G. with two vertices corresponding to edges [a, b] and [x, y] of G joined together
'/. and only if, { a, b, x, y } is not a clique In G. The minimum order Q(G) of the
^ypergraphs for which G Is a representative graph equals the chromatic number
G

02
The graph G corresponding to graph G in Fig. 17.2 is shown in Fig. 174
ob he —
0
1. We shall show that each (/-colouring ($,.3a Sv) of G with qm
y(G) colours yields n hypcrgrnph (£; *,. Xa Xn) of order q for which
graph G is the representative graph.
The set 5, of vertices of G with the /-th colour is a stable set. If [a, h] is an
edge of G belonging to £,. then vertex a is cither identical or joined to each
of the endpoints of nn edge of S,. Therefore, the end points of the edges in 5,
generate a clique E, of G.
The hypcrgraph (X: Ev, £a Ev) hns the property that each edge and
each vertex of G arc covered by at least one of the E,. From Proposition I, C
is the representative graph of the dual hypcrgraph (£; Xu Xt ,Vn) which
has order \E\ - q. Hence.
Q(G) < y (G).
2. We shall show that a hypcrgraph // - (E; *,, Xa Xn) of order
q - Q(G) for which G - L(H) can determine (/-colouring (Si, Sa Sq) of
the vertices of £
Let A" denote the set of vertices of // that belong to exactly k of the sets
*,.Thcn.
To each vertex e A1, associate the I -clique {*„„} where e e ,V,W). To each
vertex e*Aat that belongs to exactly two sets Xuti and Xm, associate the
h*
2-cliquc {xIUi, .vy,,,} of G. To each vertex e e A3 that belongs to exactly
of the sets *,<„, Xm, A\,,,. associate the 3-cliquc {.v((,,. xMi, .vw,>} of
C, etc. Since // is of minimum order, we have A1 — 0.
In this way. a family (£1. £a £„) of (/-cliques in G is defined. Clearly.
each edge [x{, x,] of G is covered by at least one of these cliques (because
X, r\ X, contains a vertex of £). Let 5t denote the set of edges of G contained
in clique Ev. Let Sa denote the set of edges of G covered by clique fa l^al

HYPBROKAPHS AND THEIR DUALS 403
i(e not covered by El. Let S3 denote the set or edges of G covered by clique
£s thnt are not covered by either Ex or £a, etc. The fnmily (SltSa
5) clearly is a ^-colouring of the vertices of 17.
'Hence. y(G) < Q(G), and so y(G) - fl(G).
Q.E.D.
Proposition 3. Let G be a simple graph without isolated vertices and without
triangles and with m edges. The minimum order of the hypvrgraphs II for which
Qm!(ll)isQ(G) -#».
Grnph Gdefined as in Proposition 2 is an /n-cliquc. Therefore, the required
minimum order is Si(G) - m.
Q.E.D.
Proposition 4 (Frdos, Goodman, P6sn [1966]). Ij G is a simple graph of
order n without isolated vertices, then
i2{G) <
[*]■ •
For each n, this bound is the best possible.
1. From Theorem E. Ch. 11), the edges nnd vertices of G can be covered
by i family of 2-cliqucs nnd 3-cliqucs (£i. £a En) where k < f — 1. From
Proposition I. G is the representative graph of the dual of hypcrgraph
(X; £,, £a L\). Since this dual has order k, the minimum required order
is
2. We shall show that forench n. there exists n grnph G of order n such that
Ifn - 2 k is even, let 6' be the complete-bipartite graph A*.*. Since grnph
c contnins neither isolated vertices nor triangles, Proposition 3 can be ap-
P'icd to G. Thus.
If/> - 2 A- + Ms odd. then let G be the complete-bipartite grnph Ai.ik+1
Proposition 3,
h - 1 n + I n1 1 r n21
- kik + I) - y- • —,— - 4- - 4 " I 4 J ■
Q.E.D.

404 HYPFRORAPHS
The following proposition characterizes the representative graphs of t|.
maximnl cliques of a graph.
Proposition 5 (Roberts, Spencer [ 1969]). A simple graph G is the representa.
live graph of the maximal cliques of some graph if, and only if, there exists in c
family of cliques (£,//€/) such that
A) each edge ofG is covered by an £,,
B) (£, / / e /) satisfies the Helly property.
Clearly, we may assume th.it G contains no isolated vertices.
1. Let G be a representative graph of the maximal cliques Xx, Xa xn
of some graph. These maximal cliques determine a conformal hypcrgraph
//-(£; V,.*a Xn). where
£ - U *< ■
Let £], Ea, ... denote the edges of the dual hypcrgraph //*. By Proposition
I, these arc cliques of G and satisfy condition (I); since // is conformal, they
also satisfy condition B), by Theorem .1.
2. Conversely, consider a family of cliques (EJiel) in G that satisfy
conditions (I) and B). Cliques E't - {.v,} for / - 1,2 n can be added to
this family without violating conditions (I) and B).
Clearly. //* - (X; £,. Ea El. E'2 F'H) is a hypcrgraph. Let
//-(£:*,.*! Xn)
be its dual. Condition (I) implies that G is a representative graph of // by
Proposition I. Condition B) implies that // is a conformal hypcrgraph by
Theorem 3. Moreover, edge X, contains vertex e\ because x,eE! — {*<}«
and this is the only edge of // that contains vertex c,'. Hence, the edges of H
arc all maximal edges.
Thus. // is the family of maximal cliques of graph (//K.
Q.E.D.
To show that graph Gin Fig. 17.2 is the representative graph of the maximal
family of cliques of a graph (namely. //.,). it suffices to consider Et ■ { w. h. c)
and Ea - { c, d, e} as the family of cliques satisfying conditions (I) and B)-
The following result which is a new generalization of a result of Krauz.
characterizes the representative graph of the A-cliqucs of a graph.
Proposition 6. A simple graph is the representative graph of the k-clitiues oj

HYPERORAPHS AND THI1R DUALS 405
$oine graph if, and only If, there exists a family (£,//€/) of cliques in G such
that
A) each edge ofG Is covered by an E,.
B) each vertex ofG h 'longs to exactly k of the £,,
C) ach partial family (E,fJ J) formed from k of the cliques E, that are
pairwise intersecting, has an intersection of cardinality one.
1, Let G be the representative graph of the family rfi\ - (A',. Xu XH)
of ^--cliques of a graph (E, f); assume that each vertex or edge that docs not
belong to a A--cliquc has been removed from (/;'. /'). Family' * determines a
uniform hypcrgrnph // - {£', Xt, X Xk) of rank k, and
G - («'),.
Let £i,/fa,... denote the edges of the dual hypcrgmph //*. Since
C * (//*)a> these edges arc cliques of G and satisfy condition (I). They also
satisfy condition B) because an edge of // contains k vertices, and thus a ver-
vertex of //• belongs to k edges. Condition C) is also satisfied because if A" vcr-
liccs e, of H arc pairwisc adjacent, then there exists a unique edge of // that
contains all k vertices (since // is con formal).
2. Conversely, let (E, / /€/) he a family of cliques of G that satisfy con-
conditions (I). B) and C), and let //♦ - (X: £,. Ea £„) be the hypergraph
whose edges arc these sets.
lly condition (I). G - (//*J. By condition B). each edge of the dual
hypergraph (//*)* — // contains exactly k vertices, l-inally. the ^-cliques of
graph (//)a arc the edges of //. and by condition C). graph Ui)t contains no
other ^-cliques. Thus. G is the representative graph of the A-cliques of graph
(")■,.
Q.l-.D.
Corollary (Krausz [1943]). A simple graph G is the representative graph of
none simple graph H if, and only if, G contains a family of cliques (/:, / iel)
«ich that
(I') each edge ofG belongs to exactly one E,,
B') each vertex ofG belongs to exactly two E,.
Let k - 2 in Proposition 6: then conditions (I) and C) imply condition
(!'). and vice versa, and condition B) becomes condition B').
Q.E.D.
Note that this corollary gives only a global characterization of an /.(//). We
now present a local characterization that is easier to verify: first some
new definitions arc required.

406
HYPriWRAPIIS
I
Tv c,
a
♦ X
M«. 17.5
A tri<inglc of n graph G ■ (\\ t:) is defined to l>c normal if each vertex
of the graph is adjacent to an even number of vertices of the trinngic. Other-
Otherwise, the triangle is defined to be special. Note that a normal triunglc is a
maximal clique of G.
I emma. ifU possesses two special triangles ahc ami ahd with [r, d] E, then
G contains as a subgraph at least one of the graphs shown in h'ig. 17.5.
We shall consider separately the following cases.
Case I. There exists a vertex ofG that is adjacent to an odd number of ver-
vertices of triangle ahc and to an odd number of vertices of triangle abd. Let x be
this vertex. Two subcases must be considered:
Case I'. Vertex x is adjacent to exactly one nrtex of each triangle. If* |S
adjacent to a, then subgraph C, is obtained with vertices x.c.iha. If* '*
adjacent to c and to </. then Gu is obtained.
Cash I". Vertex x is adjacent to all the vertices of one oj the two triangles-
Then, x must be adjacent to «, b. c. d. and Gt is obtained.
Cash 2. Each vertex ofG Is adjacent to an even number of ten ices in e>tner
one of the triangles ahc or abd.
Let x be a vertex of G adjacent to an odd number of vertices of aba let y "*
a vertex adjacent to an odd number of vertices of ahd. Thus, x y.

IIYPI RGRAI'HS AND Till IR DUALS 407
First note that .v is not adjacent to both c and d (because, otherwise, x
would he adjacent to a, />. c, d and then the conditions for Case 2 would not
tic satisfied). If .v is not adjacent to c. then x is adjacent to exactly one of the
vertices a or />. and consequently, to d.
rims, three subcases must be considered.
I. x is adjacent to a, b, c but not to </.
II. x is adjacent to c. but not to a, b, d,
III. .v is adjacent to a, d, but not to b, c.
By considering the conditions on x and y and whether or not x and y arc
adjacent, one of the nine graphs of Fig. 17.5 can be obtained. The details arc
left to the render.
Q.E.D.
Theorem 4 (Beinekc [1968]). For a simple graph G, the following conditions
ore equivalent:
1I) G is the representative graph of a simple graph,
B) G contains none of the subgraphs shown in Fig. 17.5,
C) the family '(> of all the maximal cliques ofC that are not normal triangles
covers each vertex at most twice, and covers each edge at most once, and
G contains no subgraph oj the type Gt,
D) there exists a family ■ ofcliqms such that each vertex belongs to exactly
tnv cliques of. and each edge belongs to exactly one clique oj. .
A) » B) From the Krauz theorem, there exists a partition of the edges of G
into cliques such that each vertex of G is covered by at most two
cliques. If G contains a subgraph of the form G,. then this partition
induces in G, a partition of the edges of C, with the same property.
It is easily verified that none of the graphs C/, of l-'ig. 17.5 admits
such a partition.
B) ■- C) If two maximal cliques C and D which arc not normal triangles
have a common edge [a, b), then there exist two vertices ceCand
de I) that arc not adjacent. If C { a, b, c}, then the triangle abc
is special. If C - {a,h,c}, then the triangle abc is also special.
From the lemma, the special triangles abc and abtl with [c, d) i E,
yield that there exists a subgraph of G of one of the types shown in
in I'ig. 17.5. This contradicts condition B).
C) - D) Let '* be the family of all the maximal cliques that are not normal
triangles. We shall successively add new cliques to this family to
form family J*.

408 IIYPrKOKAPHS
Let [a, b\ be an edge th.it is not covered by a clique of , This edge necessarily
belongs to a normal triangle abc.
No clique of covers only one vertex of this triangle (because, otherwise,
the triangle would be spcci.il). Similarly, no clique of V covers the three
vertices of this triangle. Two cases must he considered:
Cask I. No vertex of triangle abc is covered by a clique of . There arc two
subcases to consider.
Case 1'. Triangle abc intersects with no other normal triangle. Form by
adding the cliques {«,/>, c},{a },{/>} and {c} (or, by adding the cliques
{a,b},{a,c),{l>.c}).
Case 1". Triangle abc has a common edge with another normal triangle (and
with only one, because G, is forbidden). Let abd be this new triangle. No other
normal triangle meets abed (because then it would intersect each triangle at
two vertices). Triangle abd does not meet any clique of (because then a or$
would be covered by a clique of V). Form by adding the cliques {a, b, c),
Case I' Case 2'
FiK. 17.fi
Case 2. A vertex of triangle abc is covered by a clique Cef . In this case,
exactly two of the vertices arc covered by clique C (because, otherwise, the
triangle abc would not be normal). Without loss of generality, let b and c be
these two vertices.
CASE 2'. No other clique of meets triangle abc. Form by adding the
cliques {a, h} and {a, c}.
Case 2". One other clique C of meets triangle abc. In this case, C meets
the triangle at two vertices (otherwise, abc would not be normal), and these
two vertices must be a and c from the definition of edge [«, />]. No other

HYPEKORAPHS AND THEIR DUALS
409
clique of meets triangle abe. Thus, form by adding the clique {a, b}. See
Fig- 17.6.
D) => (I) by the corollary to Proposition 6.
Q.E.D.
a
7H
Flu. 17.7
Remark. Note that Theorem 4 presents an efficient method to determine
if a graph G is the representative graph of some simple graph. An ex-
example is given in Fig. 17.7. The graph // represented by (! can be constructed
from the family V of the maximal cliques of G that arc not normal triangles
(and which are circled by a continuous line in Fig. 17.7). In graph G. the only
normal triangles arc 234,456. and 678, which arc not to be included in family
#. Family can be formed from ', by adding the 2-cliquc { 7, 8 } and the
I-clique { I }. Hence, the required gr.iph //, shown in Fig. 17.7, is obtuincd.
Proposition 7. A simple graph G is the representatire graph of some multi-
graph without triangles and without loops if, and otdy if, each vertex of G be-
belongs to at most two maximal cliques.
I. Suppose that G » L(H) is the representative graph of multigraph //. A
vcrtex * of G represents an edge [e, d) of //, and I'^x) U {.v} represents the
set of edges of // incident to r or to d. Let Cx denote the set of vertices corre-
corresponding to the edges incident to c, and let Dx denote the set of vertices
corresponding to the edges incident to d. Then
u { x
- Cx u Dx
a|id Cx and Dx are two cliques of G. Since // contains no triangles, the sets
Cx - Dx and Dx - Cx arc not joined by an edge.

410 HYPERORAPHS
Let C be a maximal clique in G that contains x. Then, C <= Cx\j Dx, \f
yeCand yeCx - Dx, then each vertexy' eC that is adjacent toy belong]
to Cx; hence, C c £,, and it follows from the maximality of C that C«f
Therefore, each maximal clique that contains x equals cither Cx or D,, and
consequently, there are at most two maximal cliques that contain x.
2. Conversely, let A" denote the set of vertices in G that belong to exactly
one maximal clique, and let A" denote the set of vertices in G that belong to
exactly two muximal cliques. Assume that X - A" U A".
Consider the hypcrgraph //• - (Af; £,, £a {x\}, {xa},...) whose
edges arc the maximal cliques of G, and the cliques {x\} for all x\ A".
Consider the hypcrgraph H - (£; AT,, Afa Xn) dual of //*. Clearly,
graph C is the representative graph of //. Hypcrgraph // is a multigraph with,
out loops, because each vertex x, of //* belongs to exactly two distinct edges.
Suppose that // contains a triangle f i e2 e3; let x,y,zbe the vertices of H*
that represent respectively edges [eu ra], [ra, ea), [c3, fi]. Then, l
x € £, n £a; >■ e £2 n £3 ; z 6 E3 n £, ;
xtEi yjE, ziEt.
Let C be u maximal clique of C that contains triangle xyz. Then,
C*£,,£,,£3.
Consequently, vertex .v belongs to three distinct maximal cliques Eit £a, C,
which is a contradiction.
Q.E.D.
Proposition 8. A simple graph G Is the rcprexennitire graph of some bipartite
mult {graph if. ami only If, the two following eontlitims hold:
A) each vertex ofG belongs to at most two maximal cliques,
B) each odd elementary cycle ofG contains two sides of a triangle.
1. If G -/(//) is the representative graph of a bipartite multigraph H.
then condition (I) follows from Proposition 7. Condition B) also holds, be-
because if [.V|, xa,.... .Vp, .v^ is an elementary odd cycle of G that docs not
contain two sides of a triangle, then /> > 3. and the corresponding edges
A",. X,,.... Xv, Xi of multigraph // also constitute an odd elementary cycle-
This contradicts that // is bipartite.
2. Conversely, consider a graph G that satisfies conditions (I) and B)-
From condition A), there exists a multigraph //-(£; A\. A"a/.... -*»)
without triangles such that G - UH). It remains to show that multigraph "
contains no odd elementary cycle [f,,f er, f|] with /> > 3.
Suppose that // contains such a cycle. I ct A", be an edge of multigraph H

HYPERGRAPHS AND THtIR DUALS 411
that joins e, and e, + l in the cycle. The sequence [xpt Jt,,;ca xp) is an
elementary odd cycle of C. From B), there exists a triangle formed from
three consecutive vertices of this cycle, say .V|, xa, x3. In multigraph H, edges
Xi, A'a, Xa are pairwisc adjacent and, since // contains no triangles, they
have a common endpoint. But this contradicts that these three edges lie on
an elementary cycle.
Q.E.D.
This theorem shows why graph C shown in Fig. 17.2 is the representative
graph of a bipartite multigraph //.,.
Remark. Consider a graph G\ that is a triangle, and a graph Ga that
is three edges with a common endpoint. Clearly. HGY) and L(G3) are both
isomorphic to a triangle. H. Whitney [1932] has shown: Ij two con-
meted simple graphs have isoniorphic representative graphs, then they are iso-
isomorphic, except for Gx and Ga.
This result has been extended to hypcrgraphs by Bcrgc and Rado [1972].
Two hypcrgraphs //-(*:£,,£„ £J and W-(Y\FltF3 Fm)
arc said to be isomorphic if they have the same number m of edges, and if there
exists a bijection^: X-*- Vand a permutation n on M ■ { 1, 2 m } such
that
<p(E,) - £,„, (/- 1.2 m).
H and //' are said to be strongly isoniorphic if there exists a bijection
ip: X-*- Vsuch that
<p(£,)- F, (/- 1.2 hi).
In this case, we write // 2 //'.
Clearly, two simple graphs (E,jieM) and (FJieM) luve the same
representative graph if, and only if, (£,. £,) 2 (F,, F,) for all /,/
The main results for hypcrgraphs are the following:
I. For m > I, there exist t\m uniform hypcrgraphs A'P(A/) - (£, / leM)
and Lm(M) - (/•', / i e M) which are not strongly isomorphic and which satisfy
(EJieM - {*}) Z iFJieM - {k)) (keM)
Tlie vertices of Kn(\l) are points x,. for all. I <= M with \./ | ■ m modulo 2. and
its edges arc the sets E, — { xf / / 6./ }for all i < m. The wrtices ofLn(M) are
Pointsy, for all.I c M with \./ | ■ m + 1 modulo 2. audits eilgesare the sets
I. Let p be an integer, p > I. /> < »/; let fl - (A": £,. £a £n) and
- ( >': Fi, Fa Fn) be two hypergraphs such that for each set J <= M
rdinality p. the partialhypergraphs (£, I icj) and{F, (teJ)are strongly

412 HYPERORAPHS
isomorphic. Then, H and H' are strongly isomorphic, except if there exists if,re
sets A<= X,B<= Y,f<=M with \l\-pand
M n E, I i I) 2 *,(/). (BnFJ /e /) s
or Nee versa.
For the proofs, see Bcrgc, Rndo [1972].
EXERCISES
1. A hypcrgruph (X, 6) \s said to be hereditary if
B<=A. B e •
Let. be the set of all subsets of of the kind < £„. £„,.... £„ } with £„ c £„ r
«= £.,. ' *
Show that the derived hypergruph (<?.■*') is conformal. (In ulgcbraic topology, a
hereditary hypcrgruph is an abstract limplicial complex, and the derived hypcrgraph
is its first barycentric subdivision.)
2. Let *i. *j, .... em be points on n line, and let Xi, X2 X» be intervals such that
no interval is contained in another. Show that the dual hypcrgraph //* of // m (A\,
Xt Xn) is also a family of intervals.
Also, show how to place the points Xi.xa,..., Xn and the intervals £|, £a,.... F«on
a line such that et c X, if, and only if, *,«£,.
3. Let C be a graph such thnt the maximum cardinality <»(G) of the cliques of C is < 3.
Show that C is the representative graph of the mnximal cliques of some graph if, and
only If, the maximal cliques of C satisfy the Hclly property.
Show that C is the representative graph of the maximnl cliques of some graph if. and
only if, G contains no partial subgraph with six vertices a, b, c, d, e,/and with edges
ab. ac, be, bd. ft*, ce, ef. de. ef. (Roberts, Spencer (I969|)
4. Calculate the minimum cnrdinality din) of n set A such that each graph of order n
is the representative graph of n distinct subsets of A. Show by induction on n that
rfB) - 2,
dO) - 3.
if ">4.
(Erdfls, Goodman. Posa |I9«I>
5. Consider a family of non-empty sets of the form
{jr/jt - (*i,jr» **)eM*;<»i < X* < 6", «* < xi < /i* a" <**<** )■
Show ihoi this fnmily satisfies the Hclly property.

HYPERORAPHS AND THUR DUALS 413
. Show I hat if C is I he representative graph of KK, then
(I) C has Q vertices,
B) G is regular of degree 2(n - 2),
C) any two non-adjacent vertices of G have exactly 4 common neighbours,
D) any iwo adjacent vertices of G have exactly n - 2 common neighbours.
A. Hoffman [I960] has shown lhai this necessary condition is also sufficient excepi
for n - 8. For n - 8. there exist three graphs different from I(#f,) lhat satisfy ihese
conditions.
7. Show that if G is the representative graph of AC™,,, then
A) C has mn vertices,,
B) G is regular of degree m + n - 2.
C) any two non-adjneent vcriiccs have cxacily 2 common neighbours,
D) ihcrc are n ("'\ pairs of adjacent veriices in C lhat have cxncily m - 2 common
neighbours, and I here are m Lj pairs of adjacent vertices in G thai have exactly
n - 1 common neighbours.
Moon (I963| and Hoffman (I964| have shown lhat this necessary condiiion is also
sufficient excepi for «i - n - 4, For this case, there does exist a graph different from
UKt,t) discovered by Shrikhande [1959] that satisfies conditions (I). B), C) and D),
8. Show that a connecied graph G is ihe reprcscmative graph of a tree if. and only if.
each block of G is a clique and if no ihree blocks have a common vertex.
(Rnmuchandra Rao A969))
9. Show lhai a graph G is ihe rcprcscntaiivc graph of the blocks of some graph if. and
only if. each block of G is a clique. (Hurary [1969])

CHAPTFR 18
Transversals
1. Matching!! and c-matchings
For a hypcrgrnph // - (X. ), n family «f0 c is defined to be n matchith
if the edges of <f0 arc pairwisc disjoint. To simplify notation, we shall assumi
that // is a simple hypcrgntph, und and <f0 arc sets. Let ."" - - 0,
An alternating sequence relative to matching 0 is defined lo be a sequence
a - (/',, £,. ^a, £a, f3, .,) of distinct edges with ihc following properties: ^
A) Fi is n member of. .
B) £, is a member of 0 - { £i. ^a. •■•. £i-i} and
C) Fitl is a member of - { /■',, Fa Fx} and
Sequence a is said lo be odd (respectively, even) if its length is odd (respec-
(respectively, even).
Sequence a is said to be maximal if no more edges satisfying properties
B) or C) can be added to it.
The following result characterizes maximum matchings.
Theorem I. In a hypergraph II, a matching £0 is maximum if, anil only !/•
there exists no odd maximal allt mating sequence relative to 0 ■
This follows from Theorem B, Ch. 13) because each maximum stable
set of the representative graph L(H) of hypergraph // corresponds to *
maximum matching of //.
Q.E.D.
Note that if hypergraph // is a graph, an odd maximal alternating sequence
defines an alternating chain connecting two distinct unsnturntcd vertices.
and Theorem I reduces to Theorem (I. Ch. 7).
414

TKANSVI'KSALS 415
Consider a simple hypergraph // - (X. ), and let' c . Let
,-{£/£€', £3Jt}.
To each vertex a*, associate an integer c(.v) such that
c(x) * | SJ.
A family' <= is defined to be a ^-matching of hypergraph // if for each
vertex x.
If c(x) - 1 for every x. then a c-matching is a matching. Every subfumily
of a c-matching is a c-matching. We shall present a characterization of
maximum c-matchings, that extends Theorem I.
An alternating sequence relative to the c-matching ' is defined 10 be a
sequence a - (F,, t\, F8, £a, F3,...) of distinct edges that alternate be-
between - - ' and and have the following properties:
A) F, is a member of - fi —' .
B) Given the odd sequence a, - (F,, £,, Fa,.... /",), let' ' be the family
obtained from by removing edges Eu E2 i'(-i and by adding edges
F,, Fa,..., F,, i.e.,
1 - (<g - ot) u (tr, - V).
Then £, must belong to' — a^ und contain a vertex x with |fcj, | > c(x).
C) Given the even sequence a\ - (F,. £,, F8,.... F,. £,), let . J be the
family • - (. -a',)\J(a\- ). Edge F,., must belong to . - o\
and contain a vertex x such that | i | > c(x), and (F,, Fa F, + 1) must
be a c-matching.
An alternating sequence a is said to be maximal if no more edges can be
added toil. In fact, a is maximal if (and only if) oncoftlic following conditions
holds:
(i) '€ - . (I fails).
(ii) a — a, is mid, and ' is a c-matching.
Clearly, if' ' is not a c-matching, then there exists a vertex x with
and since (Flt Fa,..., F,) is a c-matching, we have
IW - *,),« | - I' \ I - | <*. - V), I > <■(*) - cix) - 0.
Therefore, there necessarily exists an edge £, e V - at that contains vertex
* and which can extend the sequence, by B).

416 IIYPERGRAPIIS
(iii) a m a\ is odd, and family . ' is a c-matching.
(iiii) a ■ a\ is odd, and family . ' is not a c-matching, but for each set
f,,i6/ - a that contains a vertex x with | * I > c(x) (which
necessarily exists), family (Ft. F2 F,¥l) is not a c-matching.
We shall now show that a c-matching'- is maximum if. and only if, there
exists no odd maximal alternating sequence.
Lemma. Let he a hypergraph, and let. and' be two disjoint c-matchlngs\
let T. m - ,~m -' , m ~~ u<T. I'lieiu "" and ?. an two c-match-
ings of hypergraph , and each alternating sequence
IT m(B,,Ct. fl2. C,,...)
of 8 relative to c-matching $ is an alternating sequence of relative to the
c-matching ' . Bt sides, if a is an odd maximal alternating sequence in , theit
a is an odd maximal alternating sequence in rf1.
The proof is easy and is left to the reader.
Theorem 2 (Ray-Chaudhuri [I960]). For a hypergraph H - {X, ), a
c-matching is maximum if. and only if, there exists no odd maximal alter-
alternating sequence relative to the c-matching .
1. Suppose that there exists an odd maximal alternating sequence a. Then
' is not a maximum matching, because the family
'- (' - ff)u(ff-tf),
is also a c-matching, and contains one more edge lhanff. (Otherwise, sequence
a is not maximal, from B).)
2. Let' be a c-matching for which there exists no odd maximal alternating
sequence, and suppose that is not maximum. We shall show that this
leads to a contradiction.
Consider the mnximum c-matching that minimizes the "distance"
Then, d(W, ) > 0 (because otherwise, ' - , and c-matching is
maximum). Let <? - ' - Si and . - Si - , and consider the hypergraph
g m. ~ul.

TRANSVERSALS 417
If we let d(x) - c(x) - | ( nV)x\, then # and ~~ are two rf-matchings
of with G n ~3i - 0 and | ~3i \ > | ff |. Moreover, there exists a Bi s 18
(because | | > | | > a). Form from B% a maximal alternating sequence
g m (Bi, Clt Ba, Ca,...) in hypcrgraph relative to the ^/-matching?.
Sequence a is even (otherwise, from the lemma, a would be an odd maxi-
maximal alternating sequence in hypcrgraph H relative to ). Thus, sequence a
is of the form:
ff-(fl,,C,,flj Bp, Cp).
There exists a set Bprl e3J - (a n 18), because 118 \ > \ % |. Furthermore,
each such set flp+1 forms a cZ-matching (£,, Bit.... Bp, flP+i) of . Since
sequence a is even and maximal, and since condition (iiii) for the maximnlity
of an alternating sequence does not apply, it follows that condition (iii)
liolds. Thus,
is a </-matchingof . Let
Then,
| | (« n V)x | - c(x).
Therefore,.»' is a c-matching of <f, and since | ' | ■ I l> family ' is a
maximum c-matching. But then. dffG, St) < d( , ), which contradicts the
definition of .
Q.E.D.
The deficiency of a c-matching is defined to be the integer
«' ) - I (c(x) - I , I) > 0 .
x*X
If f(x) m 1. then SCX) equals the number of unsaturatcd vertices. If
hypcrgraph // is uniform of rank //. then a maximum matching ' is also a
matching of minimum deficiency. The following theorem shows how the
determination of a c-matching with zero deficiency can reduce to the deter-
determination of a matching with zero deficiency.

418
HYPERORAPHS
H
H
Fig. 18.1
1 hcorcm 3. Construct a hypergraph FI corresponding to hypergrapii H -
(X, 6) as follows: For each x s X, consider two disjoint sets
-4*- {"{
- 1.2 \gx\-c(x)}.
and let
X - U Ax u U Bx .
xmX xtX
Let the elements of -F be the vertices of FI, and let each set of the form
and each set of the form {«{,./>*} />e an edge of (sec Fig. 18,1).
Tlien. to tach matching % of FI that saturates all of the /£. there corre-
corresponds in H a C'lnatching' , and vice versa
Furthermore, to each matching ? of with zero deficiency, there corre-
corresponds in 11 a t-matching V with zero deficiency, and vice versa.
1. If is a matching in 77 lhat saturates every />J, then
Thus.' is n c-matching in //.

TKAN3VKKSAL5 419
Conversely, if is a c-matching in //, then induces in /7 a matching
that can easily be completed so that it saturates every b"x.
2. Let Ax denote the set of vertices in AK that arc saturated by mulching
#, Then.
- I (-*(*) +|<f,| + |/,|)- I \*.\
x»X xtX
- IIBJ+ I l<f,l- I I'M - I Mil- I l*«l-
x.X *.X x«X Ji.X x,X
Therefore, '< is a c-matching with zero deficiency if, and only if. # is a
ni.itching of // with zero deficiency.
Q.E.D.
Given a non-ncgntive integer d(x) for each jc € X. a family c j is said
to be a A-corering if
| x\>d(x) {xeX).
If d{x) - I for each vertex .v, then a d-covcring is a covering in the usual
sense. The following theorem shows that u minimum d-covcring problem
reduces to a maximum c-matchir.g problem.
theorem 4 (Kay-Chaudhuri [I960]). Consider a hypergraph with vertex
numbers c(x) and rf(.v) > 0 .vi«-/i that
c{x) + d(x)-\Sx\ {xeX).
A family of edges is a maximum c-ntatching if, and only if, the family
& ■ — ' is a minimum i-covering.
Note that \' x\ < c(.v) is equivalent to
Thus, is a c-matching if. and only if.. is a d-covcring.
Let 0 be a maximum c-matching and let . 0 be a minimum d-covcring:
then
Since 0 is maximum, the c-matching — 0 is also maximum. Thus,
(lie above inequality becomes an equality, and | — o I ™ I o I- Thus.
- ' o is a minimum d-covering.
Q.E.D.

420 HYPERORAPHS
2. Transversal number
A transversal of a hypcrgraph // - (A': £\. Ea Em) is defined to be a
set T c A' such that
TnE,t 0 (/- 1.2 w).
The transversal number is defined to be the minimum number of vertices in
a transversal; the transversal number of hypcrgraph // is denoted by
t(//)- min| T\.
Many combinatorial problems involve the calculation of the transversal
number. V
Exampie I. Determination of a maximum stable set in a graph. Let
G - (A'. E) be a simple graph. From Chapter 13. we know that a set S c Jy
is stable if, and only if, each edge of G has at least one of its (Midpoints in
X - S. i.e., if, and only if, T - A' - 5 is a transversal of graph G. Thus,
the determination of a maximum stable set reduces to the determination of
a minimum transversal.
Example 2. Determination of the minimum absorbant set in a \-graph
G - (A', f), Consider a hypcrgraph // on A' whose edges arc
Ex - { x } u r(x).
Each absorbant set of G is a transversal of //. and vice versa. Hence the
determination of a minimum absorbant set reduces to the determination of
a minimum transversal,
Dual to the minimum transversal problem is the minimum covering prob-
problem. For a hypcrgraph // - (A', rf) a covering is a family . of edges whose
union is A'. The cardinality of a minimum covering of // is denoted by p(H).
Since each transversal of // corresponds to a covering of the dual //*«
and vice versa, it follows that p(ll) - t(//*).
Example. Quine's problem (switching theory).
Consider the boolean variables z,, z zn (that take values 0 and 1).
and a function «)(:,,:, rn) defined on [0. l]m, that takes only the values
0 and 1. A boolean sum. boolean product and boolean complement arc defined
as follows:

TRANSVERSALS
421
'I:
I:
0
i
0
i
0
1
if z - z'
otherwise,
if
if
if
if
z -0
Z'Z'
z - 1
z -0.
-0
or z'
- 1 .
-0
Clearly, <p can be written as a polynomial in z,, z zm. J,. 73>..., fm.
Quinc's problem is to express <p as the sum of the smallest possible number
of monomials,
To show how this problem reduces to the minimum covering problem,
consider a hypergraph // whose vertices arc the 2m vertices of an /n-dimcn-
sional hypercube and whose edges arc the /(-faces of this hypercube for
k - 0. 1.2 m, For example, consider a function tp of 4 variables whose
non-zero values arc given in the "truth table" below.
0
1
1
1
1
1
0
1
1
1
1
1
1
0
1
1
0
1
0
1
0
1
0
0
1
0
1
0
1
1
1
0
0
0
1
1
1
0
0
1
1
0
0
1
1
1
1
0
1
1
<p(z,,z2.zs,z4) -
Let A be the 10 vertices of the hypercube that correspond to the non-zero
values of function <p (circled in Fig. 18.2), The vertices of//,, can be covered
0111
mi
0011
0001
1001
Fig. 18.2

422 HYPKKOKAPIIS
by exactly 5 edges of HA (which urc hutched in Fig. 18.2). Edge {0101,
corresponds to the monomial ?iZa?3 that takes the value I only ut the r
ticcs covered by this edge. Similarly, the edges {0111 }, { 1010, 1110. 1 loo
1000}.{ 1000. 1100. HOI. 1001 }.{ 1001. 1011 } correspond respectively to the
monomiuls ?iZ3z3;4. z,?,. z{lz, Zi?3z«. Hence.
<p{zt. z2, z3. z4) - z, z2 z3 z4 + z, z2 z3 + r, z2 zt + z, zA + z, 5, .
Before introducing a method to find the minimum transvcrsuls of a
hypergrnph // or, more generally, of a family of sets, some additional defini-
definitions urc needed.
Let .«•-(/!,//€/) und - (B,IJ*J) he two sets of subsets of X I
{Xi.Xa, ...*„}; let *
s4 Adf - (A,nBjl(i,j)eI x J).
• •
Let s/ (the sieve of. ) denote the set of all the transversals of . Let
Cl sf denote the set of all sets that contain ut leust one A, . Let Min.
denote the set of all minimal sets of •a'. Finully, let Tr^/ denote the set of
ull the minimal sets of sieve s/'. It follows thut
Tr s4 - Min (.<• ).
,oT- Cl Tr st .
Proposition 1. -We have J'- Cl .<•"- (d'j/).
Proposition 2. — We have Cl ^ - Cl Min ^.
Proposition 3. — We have (sf u &) - .s/n ." .
Proposition 4. - We /wre Min (Cl s4 n Cl &) - Min (^ v #).
Clearly.
Cs A for some /I e s4
v | C 3 fl for some h e 3t
— C s A<uB —
Furthermore.
D- A' u fl' s Cl ^ n Cl

TKANSVBKSALS 423
Thus. D contuins a C e Min (Cl s/ r\ Cl. ). Since C = D = C and C is
minimal, it follows thai
C- C -
Hence,
Min (Cl ^ n Cl Jf) c Min (si v ).
Since the inclusion is sutisficd in the opposite direction, the equulity follows.
Q.E.D.
Proposition 5. We have:
Tr (^ u . ) - Min (Tr s4 v Tr , ).
By successively applying Propositions 3. 1,2, 4, we have:
Tr {s/kj9)- Min {Jf n if) - Min (Cl at n Cl £)
- Min (Tr st v Tr 9). q E D
Algorithm to construct Tr. .
First, determine Min .$/-{/!,. A , Ak },
Next, successively determine the following families:
^,-{-4.} - Tt{Al)m({a}lamAl)
.s/j - .s/, u{/l2} -» Trs*! - Min(Tr^/, v Tr{^2})
^-^,u{y<,} -» Tr^,-Min (Tr. 2 v Tr{/1,}), etc.
Proposition 5 shows how Tr.«/U1 is obtained from Tr. ,. If Min.«/ has
k members, then the algorithm constructs Tr c/ - Tr.c/V in k steps.
Maghout [1959] has expressed this algorithm in terms of Boolean opera-
operations.
Hi is algorithm calculates all minimal transvcrsuls und, therefore, ull
minimum trunsvcrsul sets. If the size of the problem is too large, or if only
one minimum transversal set is required, or if only a transversal set of
cardinality less than some fixed number is required, then variations can be
employed. The reader is referred to Roy [1970] and Lawlcr [1966) for the
details.
Consider a hypcrgraph // with a transversal number t(//). If i<//) denotes
the maximum cardinality of a matching in //, then some simple relations
exist between v(//) and t(//). First, note that if /(//) is the representative
graph of //. then
V(H)mm -

424 HYPERORAPHS
Thus, the maximum matching problem reduces to a minimum transversal
problem.
Theorem 5. Let H be a hypergraph; then
Clearly, if 0 is a matching of H and if T is a transversal of //. then
Hence,
min | 7 | > max | S0 | .
Q.E.D. \
Theorem 6. Let II be a hypergraph with rank r(X) - h, then
xUI)*hv(H).
Let S - (£,//€/); let <f0 -(£,// eJ) c £ be a maximum matching. Then.
| J | - v(//), and U,#/ F, is a transversal. Therefore.
■ I I £. I < \J I KX) - r(X) v(H).
; | (U i
Q.E.D.
A hypergraph // - (A'; £„ £a,..., £n) is defined to be x-critical if
t(// - £,) < x(H) (/ - 1. 2 m).
Let /(/i. ft) denote the maximum possible number of edges in a r-critical
hypergraph of rank h with t(//) - ft + I. If // is a graph, then H is t-
critical with t(//) - ft + 1 if. and only if, // is ot-critical with ot(G) - n -
k - I. (see Chapter 13). since the complement of a maximum stable set is
a minimum transversal, and vice versa. Theorem A0. Ch. 13). can be ex-
expressed as
()
Erdfls and Gallai [1961] conjectured that for each h,f(h, ft) - I J •
The following theorem presents a proof for this conjecture.
Theorem 7 (Jaeger. Payan [1971 ]), // // - (X, ) is a t-criticai hypergraph
with rank h and with x{H) - ft + I. then the number of edges in H is smaller
than or equal to I I.

TRANSVERSALS 423
Equality holds for a hypergraph {X, ) with \ X | - A + k and 6 - „(*).
1. We may assume that // is uniform of rank A: otherwise, by adding
h - | E, | auxiliary vertices to each edge £, with | £, | < h, we obtain such
a hypergraph.
2. Let H « (AT; £|, £a,.... £m). By hypothesis, // - £, contains a trans-
transversal of cardinality k. Let F, be such a transversal.
Since t(//) - k + 1, set F, cannot be a transversal of //, but F,r\E, 0
Tor ally f* /; it follows that
Ft n £, - 0.
Let Z be the set of ordered pairs (A, B) of subsets of X such that
| M | - A. | fl | - * , Mnfl-0.
Consider the graph G whose vertex set is Z and with vertices M. B) and
(/!', fl') joined together if A r\ B' - 0 or /T n B - 0.
There exists a subset y of A' with cardinality h + k (because | X \ >
| £, u f, I - A + A), and for such a set y. let
SV -{(M. «)/(/«. B)eZ. AUB- Y}.
3. We shall show that 5r is a stable set in G. Consider two distinct vortices
.- - {A. B) and zF - {A1. B') in SY: we have
AU fl- A'\J B' - y,
M n B - M' n fl' - 0 .
Consequently, M n fl' 0 and A' (~\ B 0. which implies that z and
r' arc non-adjacent in G. Thus. SY is a stable set in G.
4. We shall show that 5V is a maximum stable set in G. Let p be a complete
ordering of X. and let C(p) denote the set of vertices (A. B)eZ such that
cipb for all a A and all be B.
C{p) is a clique in G. because if {A. B) and (A\ B') are two distinct ver-
vertices of C(p). and if A n B' 0. a vertex c in A r\ B' satisfies a' pc for
each a' e A' and c p b for each ft € B; thus l'nB-0, and vertices (/I. fl)
and (M', B') arc adjacent in C.
It is easy to sec that for each Y and each p.
C{p) r\Sr 0.
Moreover, for all zeZ. the number of cliques C(p) that contain z has a
constant value d > I. Therefore, the total number t of distinct cliques C(p)
satisfies
d\St\-t.

426 HYPBRGRAPH3
Since d \ S | < t for each stable set 5. it follows that SY is a maximum
stable set.
5. The vertices (£,, F,),^, Fa) (£m, Fm) constitute a stable set jn
C, and consequently, «(G) > m.
Since a(C) - | Sr | - ( I. it follows that
^ (h + k\
EXERCISES
Q.E.D.
{
1. Let H - (X, ) be a uniform hypcrgraph of rank A. Let <?0 be a maximum matching
in H. Let A( 0) denote the set of unsaturntcd vertices in matching <f0. For S «= AT, l«t
//(D denote the family of the £i e * such that £,nj. 0, and let />*(//) denote the num-
number of connected components of hypcrgraph // that have cardinality 1 modulo h. Show
that
I Ai*0) I > max (/>»<//,„) - (h - 1) | S |].
5x
(N. Saucr [1969] has given an additional condition that implies equality.)
2. Let A" be a set of points on a line. Let £(. Et £„ be subsets of X such that each
£, is the union of / disjoint intervals. If the intersection of nny / + I of the E, is non-
nonempty, show that there exists a transversal of the £, with cardinality r.
3. A. Gy&rfds and J. Lchcl have made the following conjecture:
"Let C be a tree, and let A,, A2 A* be sets of the vertices of G such that for each
/. the subgraph CM is a forest with 1 trees. If the intersection of any / + I of the A, ii
non-empty, then there exists a trunsvcrsul of the A, with cardinality /."
Show that this conjecture is true for / < 3.
4. Let C be a tree, and let A\. A, An be sets of vertices of C such that for each /. Gt\
is a bi-tree (a forest with two trees). Show that if the intersection of each pair of A, is non-
nonempty, then there exists a transversal of the A, with cardinality 3.
(Gydrfos. Lchel [19701)
5. Consider a set X (possibly infinite) and two families nnd M of non-empty subsets of
X. If M and .# are hereditary families (i.e. - Cl j/ and - Cl ), show that the
following properties hold:
J' "
A) J'- " Implies
B) (^') - si
C)
D)

TRANSVERSALS 427
6. If .</ is a hereditary family that is closed with respect to intersection (i.e. A,
implies An Be ). and if si does not contain the empty set. then family si is called a
filter. For example, the family j/ - Cl ( jt0 ) is a (liter, and the family of neighbourhoods
of Xo in a topological space is a filter.
Show that if is a filter, then si «= si.
7. If si is a filter, then each family with Cl * - j/ is called a basts of filter . Let
y ■< .if for each B e , there exists an A e si with A<=- B.
Show that if ■ is the basis of a filter, and if is a family such that •<* ■< and ii <st,
then V is also a basis of a filter.
9. Using Exercise 7, show that if is the basis of a filter, then the following conditions
arc equivalent:
A) for each set A «= X, either <{A}ori»<{X-A),
• • ••
B) ■< and -< ,
C) ' ■< implies .■*<'.
(A basis of filter ■ with this property is called an ultra-filter basis.)
9. IF X is the edge set of the complete bipartite graph KP,,, and if «f is the family of sets
of edges of K,,, that form a graph K.,f, then show that H — (X, ) is a hypcrgraph
with
■«'--1 Mil-[Mr]])-
(I.. W. Bcincke [1970))
10. If X is the set of edges of the complete graph *., and if <T is the set of cycles of *„.
then show that H - (X, ) is a hypcrgraph with
(Chartrand. Gellcr. Hcdetniemi [19701; Guy [19671)
II. Repeat Exercise 10 with the complete bipartite graph K,., in place of *„; show that
v<//)-[£][*] if Mis even.
(Chartrand, Gellcr. Hcdetniemi A970))
■2. Show that if // is a hypcrgraph formed from the maximal cliques of a simple graph
0', then t<//) equals the dominance number fi'<C).

CHAPTER 19
Chromatic Number of a Hypergraph
1, Stability number and chromatic number of a hypcrgraph
Given a hypcrgraph // - (X, ff), a set 5 c X is defined lo be stable if Iji
contains no edge £, with | £, | > 1. The stability number of // is defined lo be
the maximum cardinality of a stable scl of //.
The chromatic number x(H) is defined lo be the smnllcsl number of colours
needed lo colour the vertices of H such thnt no edge £, of H with | £, | > p
has all its vertices with the snme colour. This concept was introduced by
Erdosand llajnal [1966].
A q-colouring is defined lo be a partition of X into q stable sets
S,,S2 5,,
each corresponding to a colour. A hypergraph for which there exists a q-
colouring is said to be q-colourable.
II
x
Fig. 19.1
Projective plane with 7 points
Examplg. The projective plane with 7 vertices forms a uniform hypcrgraph
// of rank 3 with 7 edges aef, adg, abc,fdb,fgc, ce<l, beg. See Fig. 19.1. Set
{a, c, e,g} is a maximum stable scl. Its chromatic number is *(//) - 3, and
a 3-colouring of // is shown in Fig. 19.1.
428

CHROMATIC NUMDER OP A HYPERORAPH 429
Proposition 1 (Erdos, Hajnal [1966]). ifH is a uniform hypergraph of rank 3
ith n vertices that satisfy
\E,nEj\*l (i+j),
then each maximal stable set S satisfies
Let 5 be a maximal stable set with s vertices. Since 5 is maximal, each
x6 X - Sbelongs to an edge Exsuch that £,-(j()cj, From the hypo-
hypothesis,
|£xn£,|<l (x,yeX-Sixty).
Thus, the sets EKr\ S have cardinality 2 and are pairwise different; hence
It is easily seen that this implies
Q.E.D.
Proposition 2. Let II be a hypergraph of order n with chromatic number
and with stability number &(H)\ then
X(H)
Consider a ^-colouring (SitSa SQ) of H with q - *(//) colours.
Then.
i-i
and the first inequality follows.
Let 5 be a maximum stable set of //. Colour the vertices of S with a first
colour, and colour each of the remaining vertices with n — fi(H) additional
colours. Hence
and the second inequality follows.
Q.E.D.
Given a hypergraph // - (X, ), the degree of a vertex x is defined to be
the maximum number of edges different from {x} (hat form a partial family
(£» / jeJ) with
£,n£, -{*} UJeJii j).

430 HYPERGRAPHS
Let the degree of x be denoted by du(x). Clearly, d,,(x) «■ 0 if, and only if, lne
only edge that contains x is {x}. The following theorem shows that the
relationship between the chromatic number of a hypergruph and its degrees is
analogous to the relationship between the chromatic number of a graph and
its degrees:
Theorem 1 (Tomescu A968]). Let E,, 5a Sq) be q-colourtng of H, and
let
dk m max d,,(x).
Then, ,
X(H) < max min {k, dk + I }. *
* ««
I. We shall show that there exists an r-colouring(S[,S'2 S'r)of H with
\
mln (k.r I
s,c u s;
Si is a maximal stable set in X — (J 5,'.
i I < k
Clearly, if 5, is not a maximal stable set in X, then vertices can be added
to 5, until a maximal stable set Sx is obtained. IT St - S'x is not a maximal
stable set in X — S[. then vertices can be added to it until a maximal stable
set S'2 is formed, etc. This gives the required colouring of //.
2. Let x<t Ui*iSip &nd lety < k; then from the maximally of S, there
exists an edge E] with
E'j e Sj u { x ) ; | £} | > 1 .
Sincere £,', for ally < *. the family (EJ. £j £i) satisfies
£,' n f; - { x } (i «t ;';(.;< ft).
From the definition of </«(*). this implies that
d,,(x) > fc .
3. Let i(x) denote the index of the set Si that contains x. From part 2,
By letting A" - /(.v) - I. it follows that

CHROMATIC NUMBIK OP A HYPIJtGRAPII 431
Let ae 5V, then i{a) < A (from part I), und
i(a) < max i(x) < max (d,,(x) + 1) • d% + 1 .
Hence,
x(M) < max <(*) < max min { fc, t/j + 1 }.
Q.E.D.
Corollary I. If II is a hypergraph with /(//) ■ q + I. ourf if the hypergraph
Ho obtained by removing vertex .v0 has chromatic number x(tf0) » ff.
L ct E,, 5a 5,) be a ^-colouring of Ho ■ //.x-{««)• 'f''m(*o) < 9. 'hen
by considering the (9 + l)-colouring (.V,, Sa £„. {x0}). it follows that
Z(W) < max min { k, dk + I } < 4 ,
which contradicts *(//)-</+ 1.
Q.E.D.
Corollary 2. #"g /.» o positive integer such that
\{xlxeX,
then
Index the vertices such that
Then, for k > q,
min { k, d,,(xh) + I} < q .
Clearly, this inequality also holds for A < q.
For the n-colouring ({ x,}, (xa} {xA}), Theorem I yields
X(H) < max min { k, d,,(xk) + I } < q .
Q.E.D.
Corollary 3, #"// is a hypergraph of maximum degree d0. then
X(H) < d0 + I .
The proof follows from Corollary 2 with q m </, + I.
Appiication. The following theorem is due to T. S. Motzkin [1968]:

432 HYPERORAPIIS
For a simple graph G with maximum degree h, it is possible to colour the ver.
I ices with \ = I + I colours such that no cycle has only one colour.
To show this, let // - (X, ) be the hypcrgraph in which X is the set of
vertices in G that belong to at least one cycle and in which is the family or
the sets of vertices that are contained in the same cycle.
The maximum degree of hypcrgraph H is d0 < [jj. and the Motzkin
theorem follows from Corollary 3.
We can also stale:
The edges of a simple graph G can he coloured with q colours such thai each,
cycle has more than one colour when
qm max min {da(a), <lo(b)}.
To show this result, consider the hypcrgraph // whose vertices are the edges
of G that belong to al least one cycle. The degree in // of an edge ab of C
equals
d,,(ab) < min { dc(a) - 1, </<,(/>) - I} < q - 1 ,
and the result follows from Corollary 3.
2. Cliques of a hypcrRraph
Let // - (X, ) be a hypcrgraph of rank h, and let r < h. A set A c A'is
defined to be a clique of rank r if either \ A \ < r, or | A \ > r and each subset
of A with cardinality r is contiiined in at least one edge of //. Clearly, each sub-
subset of a clique of rank r is also a clique of rank r. I el cur(//) denote the maxi-
maximum cardinality of a clique of rank r in //.
Since r < h, and since an edge £j with | Et \ - h is a clique of rank r,
then
| 4 | - r(H).
If the vertex set X of hypcrgraph // is a clique of rank r, then hypcrgraph
// is said to be r-complete.
If // is a uniform hypcrgraph of rank A, then a "clique of rank A" is often
called a clique, and each set with less than h vertices is called a trivial clique-
Remark I. Saner [1971] has proved the following generalization of the
Turin theorem:

CHROMATIC NUMRbR OF A HVPFRORAPH 433
Consider a set X with \ X | - n and a partition (Alt A, Ap) of X into p
classes At with
Let //B.p - (X, )bea hypergraph in which 6 is the set of subsets E of X such
that
£| - A
. | £ n Ai | < I (/ < p).
Then, //„.,, is the only simple uniform hypergraph with «(//) - n, r(//) - A,
< p. and with the maximum number of edges.
Remark 2. Let m(». A, r) denote the smallest possible number of edges in
a uniform hypcrgruph of rank A and order n that is r-complete.
If £lt £a £„ nrc the edges of such a hypergraph, then
U ,(£.) - * ,(X),
1-1
and, hence,
"(H)-
Therefore,
„,,*,, >(;)(*)-.
Furthermore, equality holds if, and only if, there exists a uniform hyper-
graph of runk A and order n such that each subset of X with cardinality r is
contained in exactly one edge of //. The hypcrgraphs with these properties are
studied in finite geometries and are called Stelner systems.
Remark 3. Chvatal [1971] has shown how the numbers m{n,h,r) arc
related to the Turrin numbers..
Let n, k, b be integers such that I < Ar </;</>, and let T(n, k, b) denote
ihc smallest value of m such that there exists a uniform hypcrgraph of rank k
and order n with m edges whose stnbility number is smaller thnn /;. The Turdn
iheorcm (Ch. 13, §2) determines the value of T(n, 2. b). The number T{n, k, b)
■s called the Turdn number in n. k, b.
If a hypergraph // - (£,//e/) has a stability number smaller than b,
then each set B with \ B\ - b is non-stnblc nnd contnins some edge £,. Thus,

434 HYPtRORAPHS
an (« - fc)-sct X — B can always be covered by X — £, for some I, an(j
hence, the hypergraph (X - £, / ie I) of rank n — k is (n — /j
Therefore,
T(n, k,b) - m(n, n — k,n — b).
From Remark 2, it follows that
\n-b)\n-b) b\(n-b)\
(n - b) l(b - k)!
(« - k) I
n\ k \(b - k) tn
' k\(n-k)\ b\ ' \kl\k) i
This inequality, which was also discovered by Katona. Ncmctz and Siinono-
vits, yields a bound for the stability number /?(//) of a hypergraph H with n
vertices and m edges such that in in,,, | £, | - A\ v
If // is uniform of rank k, then
and. therefore.
If // is not unifonn, this inequality is satisfied a fortiori.
'theorem 2. Let // be a simple unijorm hypergraph of rank h. ami let Kbea
clique with k rerticcs, k > h. Then each vertex of K has degree
ami the chromatic number of the xubhypcrgraph generated by K is
X(ffK) - do + I •
I. The degree of a vertex .ve K is the maximum number of pairwise dis-
disjoint subsets of K with h — 1 elements that do not contain x. Thus,
"' For asummnry of previous rcsuha for 7"(/i, *, A),xec V.Chvnlnl(l971J. More recci«ly
J. Spencer [1972] found an improved lower bound for 7"(n. k, b) for lnrge values of «■
He proved that if n2bk(k-I), then
TOt,k,b) >. (A-/I"" Vc-I)"-1 «**-*.

CHKOMA1IC NUMBER Of A HYPIRORAI'H
2. A minimum ^-colouring of Hk is oblnined from a partition
435
such that
Therefore,
l;rom that part 1. it follows that
\S,\-h- 1 (/- 1.2 q- I)
I 5, | < h - I
l-'urthcrmorc, from Corollury 3 to Tlicorcm I, x(MJ < d0 + 1. Hence.
Q.E.D.
Remark I. Theorem 2 shows that the bound given by Corollary 3 to
Theorem 1 is the best possible: If // is a uniform hypergraph of rank h and
maximum degree d0. then #(//) < d0 + 1, and equality holds for each //
that contains a clique whose vertices have degree d0.
However. #(//) - d0 + 1 docs not imply the existence of such a clique, as
shown by the hypergraph // in Fig. 19.2 due to L. Lovdsz. The edges of //
urc
abc,,
, abc3, c, fj c3, acb,, aibJt
fc,fcjfc3, bca\, bcat, bcat, axa1ay
FIr. 19.2

436 HYPfRORAPHS
Hypergraph // is uniform of rank 3 and contains only vertices with
degree d0 - 2. The chromatic number of H is 3 - d0 + I. However, H con-
tains no cliques of degree d0 - 2.
The following result due to Lovasz [1968) generalizes Theorem F, Ch. 15).
If H Is a connected uniform hypergraph of rank h > 1 with maximum degree
d0 > I, and if \ \Jh,x F. | < </,,(/» - 1) + I for each x. then >;(//) - d0 + i
holds only in the two following cases:
A) h - d0 - 2, and II is an odd cycle, or
B) // contains a clique of degree d0.
Remark 2. If // is a uniform hypergraph of rank h with a maximum clique
K of cardinality k. then from Theorem 2. ;;(//) is bounded below by a function
of h and k. However, /(//) is not bounded above by a function of /i and k.
In fact, Erd5s nnd Hajnul [1966] have shown that for each 9 > I, there exists
n uniform hypcrgruph // of rank h with /(//) - q (hut contains no cliques of
order > h. More precisely, *
Ifh. q. I are integers > 1. there exists a uniform hypergraph II'{h, q, I) of rank
h with chromatic number q and without cycles of length < L
A constructive proof h is been given by Lovdsz [1968].
The following is n well known result:
iheorcm 3 (Ramsey [1930]). Consider three integers p, q, h with p,q>h.
There exists afinite integer R*(p, </) such that each uniform hypergraph of rank
h with n vertices, n > Rh(p. q). contains a clique with p vertices or a stable set
with q vertices. The "Ramsey number of rank h in p and q" is defined to be
th1 smallest number Rn{p,q) with this property.
We shnll show by induction on h that if | A' | is sufficiently Inrge, then for
any purtition of *{X) into two classes S and . , there exists a set A c X
with I A I - p, n(/4) c ., or there exists a set B <=■ X with
\B\-q, h(B) c IF.
1. Clearly, the theorem is true for h - I, because if | X \ - p + q - '•
and if X is partitioned into two classes, with no p elements of X in the first
class, then there exist at least q elements of X in the second class. Therefore,
A) Ri(p. q) - p + q- I.
We shall assume that the theorem is true for ranks less than h, and «*
shall show that it is true for the rank h > 1.
2. Clearly, the theorem is true for p - h because if | A' | > q and if X con-
contains no /i-sct in , then each A-set belongs to . Therefore,
B) R,(h, q)-q.

CHROMATIC NUMDER OF A HVPI'KOKAPH 437
By the same argument, the theorem is true for q - h. and
B') Rk(p, h)-p
Since the theorem is true for p + q - 2 /», we shall prove it by induction
on p + q.
3. Let p, q be two integers with p + q > 2 h; we shnll show that
C) **(/>. q) < *»-,[/«/» -!.</). **(/>. ? - D] + I .
Let
p1- Rh(p-\.q), q'- Rh(p. q - \).
ond consider a set A' with cardinality n > /?„.,(/>', </') + I.
It sufTiccs to show that for any partition ( , )of h{X). there exists either
a set A with \ A \ - p and . h{A) c gt or a set B with \ B \ - q and
Let a e X, and let A" - X - {a}. For £' 6 0\-,(A"). let
i £'6^' if £'u{fl}e .
Since | A" | > R^-dp'. ?'). there exists (for example) a set A' <= A" with
Since | A' | - p' - fl,,(/> - 1,9). there exists cither a set B" <=■ A' with
I B" I - q . W) c: ^
(in which case the above proposition is true), or a set A ' <=■ A' with
I ^4" I - p - I, &AA") c g.
In this case, set A" u {a} satisfies
and the above proposition is true.
Q.E.D.
Corollary. Let p,q>2, and let G - (A\ E) be a simple graph of order
there exists in G either a clique of cardinality p or a stable set of cardinality

438 HYPFRORAP1IS
From Theorem 3, it suffices to show that
This is true for p - 2, since from B)
Suppose that the inequality is valid for each Raip'. </') with p + q' < p +
then from (I) and C), it follows that
R2(p. <l) < Ri(p - \,q) + Rj(p. 1 - I) - I + I
which completes the proof.
Q.ED.
Riip. q)
p + q-4
p + q- 5
p + q- 6
p + </- 7
/> + q - 8
/>-2
2
3
4
5
6
P-3
1
3
6
9
14
p-4
1
4
9
18
P-S
1
5
14
p-6
1
6
C,+!72)
p + q-4
p + q - 5
p + ?- 6
p + q-1
p + q-t
P-2
2
3
4
5
6
1
3
6
10
IS
/>-4
1
4
10
20
P-S
1 '
5
IS
1
6

CHROMATIC NUMBER OF A HYPERORAPH 439
Remark. So far, the numbers Ra(p, q) have been determined only for very
small values of p and q. These numbers are almost equal to the binomial
numbers \f "J" £ J ), (see the tables above).
Very little is known about the Ramsey numbers.
Sobczyk 11968], Kdry and Kalbfleisch have also shown that /?3C, 6) - 18;
Graver has shown that flaC, 7) - 23.
Moreover, Erdos [1947) has proved that for/> > 3,
Frasnay [1963] has proved that:
*a(/>./>)
Giraud A968J has proved that forp > 5,
p-
Application (Hcdrlfn [1966]). Let G .11 be the normal product of two
simple graphs G - (X, E) and H - {Y. F). then
H) < RJaQG) +
We have shown (Proposition 9, §6, Ch. 16) that
We shall now describe an upper bound for a(G. H); let
k - /fala(C) + l,a(W) + I].
Suppose that a(G. H) > k. Then there exists in G. //a stable set S c X x Y
with | S | - k. Let jc>\ *y e S. If jc a' and I*, x') $ E, join xy and *'/ by a
red arc; otherwise, y y' and [y, y'\ $ F, and join xy and .v'/ by a blue arc.
Since \S\ - k, there exists in the red graph a clique C with | C \ - cc(G) + I,
or there exists in the blue graph a clique D with | D \ - «(//) + I In the
first case, the set
{xjxe X, xyeC)
is a stable set of graph G with a(G) + I elements, which is a contradiction.
In the second case, we obtain the same contradiction with graph //. Hence,
//) < *.
Q.E.D.

440 HYPERORAPHS
This result, applied to the Shannon Problem (Ch. 16, §6), shows that the
graph G in Fig. 16.10, with stability number a(G) - 2, satisfies:
«(Ga) < RaQ, 3) - 1 - 5.
3. Good colourings of the edges of a graph
Consider a simple graph G - (X, £). A good colouring of the edges of C is
defined to be a mapping g from E into a given set of colours such that no
three edges of a triangle have the same colour. If the set {g(')feeE) has
cardinality q, then g is called a good q-colourlng. \
Consider a hypcrgruph whose vertices are the edges of graph G, and whose
edges are the triples of edges of G that form a triungle. Let this hypergraph be
called the triangle hypergraph of graph G and be denoted by GT. The smallest
value q for which the edges of G have u good ^-colouring equals? the chromatic
number x(GT).
Let n(q) denote the maximum order of a complete graph that has a good q-
colouring. The number n(q) is related to the Ramsey theorem. In fact, the
Rumsey numbers R,,{p, q) defined in Section 2 can be generalized to Ramsey
numbers Rh{pi.pa />«,) ofq variables with the following property:
// | X | > Kipt ,p />,), for each partition ( ,, &a,..., „) of the set
h(X) of all subsets of X with cardinality h, there exist one class , and a set
A, c X with
M/1 - Pi: PniA,) c a,.
Thus, it follows thul
r>(q) - Rjt^^) - 1 .
"V"
Proposition 1. !fn(q) is the maximum order of a clique with a good colouring
of its edges, then
Let A' be a clique of order n(q), and consider a good ^-colouring of its
edges. Let a be a vertex of A', and let A, denote the set of vertices linked to a by
an edge with the /-th colour. The clique KAl has its edges coloured with a good
(q - l)-colouring, and therefore
I A, | < n(q - 1) .

CHROMATIC NUMBER OP A HVPERGRAPH 441
Hence
n(q) - 1 - dK(a) - £ \ A, | < q.n(q - 1).
1-1
Clearly, n(l) ■ 2, und we have:
«A) < I +1
«B) < 2 «A) + 1
n(q - 1) < (9 - 1) «(9 - 2) + 1
Hence,
n(q) < 1 + q + <?(<? - 1) + ... + q ! + q ! - £ ?4
Q.E.D.
For q - I, 2, 3, the number n(</) is equul to
For 9 - 1, we huvc/(l) - 2, and since Ka has a good I-colouring. n{\) — 2.
For </ ■ 2, we have/B) * 5, and since K, has a good 2-colouring (shown in
fig. 19.3). nB) - 5. For q - 3, we havc/C) ■ 16, and K,6 has exactly two
non-isomorphic good 3-colourings respectively discovered by Greenwood
tind Gleason [19SS] und by Knlbflcisch and Stanton [1968] (sec Fig. 19.4);
thus,/C) - 16. For q - 4, we huve/D) ■ 65, but so far no good 4-colouring
has been found for KA, (see Graham [1968]); a good 4-colouring has been
found for K64 and we know that 51 < RC. 3, 3. 3) < 65 (Chung [1973]).
Colour 1
Colour 2
Flu. 19.3. Good 2-colouring of Kt

442 IIYPERUUAPIIS
Proposition 2. Let q be an integer, and let G be a graph with chromatic
number y(G") < «(<"/); llun the edges of G have u goodq-colouring.
Consider a colouring g: X-*■ { I, 2,.... n(q)} of the vertices of G in n{q)
colours. Denote the vertices of AnW) by 1.2 «(</) and consider a good q-
colouring of tlic edges of the clique A'n(v). IT edge [.v, v] of G is coloured with
the same colour as edge (#(.v), g(y)) in A'nD), then no triangle of G has all jts
edges with tlic same colour.
Q.E.D.
Proposition 3. / el G be a simple graph, and lei ax be a reriex o/G. If H /.»the
graph obtained from G by adding a vertex a0 that is foined lo each v rtex of
r() * \
1. Let q ■ /\GT). Consider a good (/-colouring of the edges of G, and
colour each new edge [n0, .v] with the same colour as edge [a{, x] in G. The
resulting colouring is a good (/-colouring of //. Hence.
XiUr) < X(Gr).
2. II </ - till1), then each good (/-colouring of the edges of //determines a
good (/-colouring of the edges of G. Consequently.
and equality holds.
Q.E.D.
Remark. A simple graph G » ( V. I') is defined to be q-ntinimal if
(!) X<G'O - q + 1 .
B) X((G-<•)')-'/ (eeE).
Clearly, each graph G' with a good (</ + 1 )-colouring has u partial subgraph
that is (/-minimal. II e shall show that A,M),, is q-minimal. Let n(q) denote the
maximum number of vertices of a complete graph with a good (/-colouring-
rhen, the graph G obtained by removing an edge "[(/„,«,] from A"b,,i4i n's0
contains a good (/-colouring from Proposition 3. Since ABWm has no good
(/-colouring, it follows that conditions (l)and B) for (/-minimality arc satisfied.
Other (/-minimal graphs have been discovered. Las Vcrgnas [1969] lias
shown that if q > 2 mid n(q) - /(</). then the graphs

16
U)
l-'ilt. IV.4. Good 3-colouring of

CHROMATIC NUMBER OF A HYPIiRGRAPH 443
(the clique with n(q) + 3 vertices less a cycle of length 5) arc the only a-
minimal graphs with less than n(q) + 4 vertices.
4. Generalizations of the chromatic number of a graph
Consider a nuiltigraph G - (X, £). A p-colouring of degree t is defined to be
a partition (Xx, X3 Xp) of X into p classes such that none of the sub-
subgraphs Gx, has maximum degree greater than r, i.e.,
maxrfo,(x) < / (/- 1,2 p).
Let y,(G) denote the smnllest integer p for which G hns n /^-colouring of
degree t. The chromatic number of G is simply yo(//), since the vertices of the
same colour determine a subgraph of maximum degree 0.
The study of y,(G) reduces to the study of the chromatic number of a
hypcrgraph // - (X,. ) whose vertex set is X and whose edges arc the sets
A c X such that GA has maximum degree greater than t. A stable set S
of hypcrgraph II - (-V,. ) generates a subgraph Gs of maximum degree
< r; thus
*(//) - y,(G).
Ihcorcm 4 (Lovdsz [1966]). lfG-(X, E) is a nuiltigraph without bops and
with maximum degree h - nux.v.x do(x), ami/flu, ha hr are twn-negatire
integers such that hx + ha +•■■+ hp ■ h — p + 1, then there exists u partition
(A,, A'j Xp) of X with
maxt/0 (x) < h, (I - 1,2 p).
xtX, '
We shall prove this result by induction on p.
I. Suppose p - 2. Since h - I - hx + h2 > 0, it follows that h > I.
For h - I, the theorem is evident. Therefore, we shall assume that h > 2.
Consider a partition (A1,, A'j) of A* that ininimi/cs
, X2) - fc,B m(GXl) -\X2\) + /.2B ;m(CXi) - | X, |) .
We shall show that the maximum degree of GXl is < ht. Let xeJf,. The
partition ( V,, ya), where Yt - -Vj - { a: }, ya - ATa u { a- }, satisfies
Since {(A^, X2) is minimum, this expression is > 0. Hence,

444 HYPBRGRAPHS
It follows that
2(A - I) ma(x, Xi) < A2 - A, + 2 AA, - (A - I) + 2 A,(A - I).
Since A - I > 0, this inequality implies that
wo(x, Xt) < r + /»| .
Since mu(x, X{) is an integer,
A similar argument shows thut
dOx (x) < A2 (x e A'j,).
Hence, the theorem is true for p - 2.
2. Let p > 2, and suppose that the theorem is true for all />' < p. Let
A'i - *i .
Thus,
Ai + A'2 - £ A, + p - 2 - (A - p + I) + p - 2 - A - 1.
Irom Part I, there exists a partition ( Yt, Y3) of X thut satisfies
max dOvi(x) < A', ,
max </0),2(x) < Ai.
Since
I A, - Ai - (p - I) + I ,
i* i
Y2 can be partitioned in (p — I) sets X2, X3 Xp with
max dCx (x) < A,,
by the induction hypothesis. Thus (Yt. X2, X3 Xp) is the required
partition.
Q.E.D.

CHROMATIC NUMBER OF A HYPEKORAPH 445
Corollary. Let G be a muhigraph without loops and with maximum degree h;
[h 1
+ I colours.
Suppose />, - I for all /. Then, there exists a ^-colouring of degree t if p
satisfies:
pt > h - p + \ ,
or
p(f + I) > A + I .
or
Thus there exists a colouring of degree / in -'- A colours, i.e.
f Jl 1+ 1 colours.
l'+lJ Q.E.D.
For / - 0. this corollary reduces to Corollary 2 to Theorem E, Ch. 15).
For t » I, this result was proved by Gcrcncscr [1965J.
A second generalisation of the chromatic number of a graph hus been
studied by G. Chartrund, D. P. Gcllcr, S. Hcdctnicmi [1968]. Let x""(G)
denote the smallest number of colours needed to colour the vertices of a
simple gruph G such that no elementary chain with length k is monocoloured
For k - I, then /n(G) is the chromatic number y(G'). For k - 2. then,
from the preceding corollary,
Z<2>(G)-yi(G) <[!■]+ 1.
No'c that /kl(G) is also the chromatic number of a uniform hypcrgraph of
rank k + I. whose edges arc the sets of vertices that lie on an elementary
chain of length k in G.
Chartrand, Gcller and Hcdctnicmi A968] have shown that if G is a simple
graph, and if the length of the longest elementary clwin in 0 is I, 2 < / < k,
then
r I 1
■2.

446 HVPVROKAPIU
A third generalization of the chromatic number of a graph can be defined
by replacing "chain" with "path" in the definition of ^
A fourth generalization of the chromatic number of a graph has been
studied by S. Hcdctnicmi [1970]. Let f(G) denote the smallest number of
colours needed to colour the vertices of graph G such thut no connected sub-
subgraph of order k has only one colour. Thus,
X2(C) - y(G),
Clearly. xk(G") is d'so the chromatic number of a hypcrgraph defined in the
obvious way.'1'
Another generalization of chromatic number has been given by H. Snchs
and M. Schaublc [1967]. For a simple graph G - (X. E) and a number
k > 2, a colouring by k-cliques is defined to be a partition (A',, A'a Xp) of
A' such that no fc-clique is contained entirely in one class of the partition
(i.e., no k-clique has only one colour). I ct y,y((i) denote the smallest integerp
for which there exists such a partition in p classes.
Thus, the chromatic number of G (in the usual sense) is Xa(G). Consider
the hypcrgraph // formed from the A*-cliques of graph G (which was charac-
characterized in Proposition 6, Ch. 17). Clearly,
Sachs and Schlkublc have discovered an inductive construction that
yields the following result:
For each p > I and for each k > 2, there exists a graph G • G(p, k) with
the following properties'.
1I) G" contains no (k + 1 Ycllques,
B) X.(«) - P.
C) for tach colouring by k-clicniex that uses p colours, there exist at least p
pairwise disjoint (k - \)-cliques that have only one colour.
(■'or k m 2. this construction gives a graph without triangles ;iik1 with
chromatic number p for each integer p (Blanche Descartes [1947J; Zykov
[1949]; Myciclski [1955]; LrckX Kudo [I960)).
A> In pnniculnr. Hcdctnicmi A9701 h.w shown thai il" V is a plimnr graph, ihcn ••'s
possible to colour il% vertices in 4 colours such Ihnt norw of the 4 subgraphs generated by
the vertices ol" Ihc same colour is connected. Furthermore, if G contains no triungles. ihc"
2 colours are sufficient.

CHROMATIC NUMH1R OF A HVPFKCKAPII 447
EXERCISES
1. Using the argument for Proposition I, Section t, show that if a uniform hypcrgraph
// with rank h > 3 satisfies
\E, nEt\ <h -2 (
then the stability number k of liypcrgrapli // snlisfics
k
2. Show that a hypcrgraph // with chronuilic number x(H) m q contains a cliuin of
length q - I. (Tomescu (I9M)])
3. Consider a hypcrgraph with n vertices and with cilges £, such that
A) |£,| - 3 for all/.
B) for a. h X, a h, there exists exactly one edge that contains both a and b.
Such a hypcrgr iph is called u Sieluer system of order n.
For example, the projective plane with 7 points (Tig. 19.1) is a Stcincr system of
order 7. Show thnt for n > 7. then &H) > 3.
It is known that a Steincr system exists if. and only if. n m I modulo ft, or if «■ 3
modulo 6. H'n ■ 3 modulo 6. a Stciner system // of order n with tf,H) - 3 can be con-
constructed by a well known method due to M. Hull (sec. Combinatorial Theory, Ginn-
lilaisilcll, Toronto. 1967, Theorem 15.3.2). If « ■ I modulo 6, n Slcincr system // with
*(//) - 3 can be constructed by a method due to A. Rosa [1970].
•i. Given an ;i x n chessbonrd, define the Queen's hypcrgraph Hi? as the hypcrgraph
whose vertices arc the squurcs of the chessboard and whose each edge £, Is the set of
squurcs that a queen at square .v controls (including square ,v>. Similarly, define the
King's hypcrgraph //„". Rook's hypergraph //", and liishop's hypcrgraph //„".
Show that
- xi'H) - 2 .
5. In the A-dimcnsional space R". let % be a family of Rn.,(.P.<l) compuct convex sets
such that each subfamily of q members, contains h + I members whose intersection is
non-empty. Show that there exist p sets of t> whose intersection is not empty.
Hint: Use Holly's theorem: If Is a family of/> > h + I convex compact sets in R".
mid if the intersection of any h + I of them is non-empty, then the intersection of the
family 't is not empty- (l)ergc. Espicrs Topologiques. Paris [1962]).
6- In the plane, let 5 be a Unite set of points with \ S\ > /?,(/>. 5). such that no three
Points arc Cm a line. Show that there exist p points of 5 that arc the vertices of a convex
polygon.
Him: Show thut
A) if | 51 - 3, then there exist 4 points of .9 that form a convex quadrilateral.
B) if all the quadrilaterals formed from a set of p points are convex, then these p points
arc the vertices of a convex polygon. (Erdos. Szckcrcs A935]).

CHAPTER 20
Balanced Hypergraphs and Unimodular Hypcrgraphs
1. Strong chromatic number
Let // » (X, <f) be u hypcrgrapli with rank function r(S). A set S is
defined to be strongly stable if r{S) » I. i.e. if
(|<] (/-I. 2 m).
Note that each strongly stable set is also a stable set.
The strong stability number <x(H) of hypergraph // is defined to be the
maximum number of vertices in a strongly stable set. Clearly.
The covering number p(//) of // is defined to be the smallest number of
edges of // that cover all the vertices of //. Clearly, p(l() > fl((//)a), the smal-
smallest number of cliques needed to cover graph (//)a. Furthermore, if // is a
conformal hypcrgraph, then p{ll) - 0((//K).
A strong q-colouring of hypcrgraph // is defined to be a ^-colouring of the
vertices of // such that no two vertices contained in the same edge have the
same colour. Clearly, a strong ^-colouring is a partition of X into a strongly
stable sets.
The strong chromatic number}'(//) of H is defined to be the smallest integer
q for which there exists a strong ^-colouring. Clearly, y(H) > r(X), Hyper-
Hypergraph // is said to be y-perfect if •/(//,,) - r{A) for all A c X.
Proposition. Every y-perfect Ity/wrgraph is conformal.
Let // » (X. <f) be a y-perfect hypcrgraph. To show that H is conformal,
it suffices to show that each clique C of graph (//K is contained in an edge
of//.
Since hypcrgraph // is y-pcrfect,
and, consequently, there exists an edge of // that contains C.
Q.E.D.
448

RAl ANCED IIYPFRURAPHS AND UNIMOIHJI AR HYPIRCiRAPIIS 449
Remark. Let 'f> denote the family of maximal cliques of graph G, If
graph G' can be characterized by a property of family '>, it is often interesting
to relate the chromatic number of G to properties of hypcrgraph // » (X, fc)
since y(G) - y(//).
The relationship between the rank of a hypcrgraph and its strong stability
number is described in the following results.
Theorem 1. Let II » (X.A) he a hypergruph with rank r(A). Then
»(//)< max [L-1JI
To demonstrate the first inequality, suppose that S is a maximum strongly
stable set. Then. r(S) - I. and
AeX
To demonstrate the second inequality, let k be the smallest number of
edges that cover X. und denote these edges by £\. £3 £i... Thus,
\A\£1.\A<\F \ 4- *•• A- \ A f\ F \ £. ItrlA\
Therefore,
kr(A) - | A | > 0 M c AT) ,
and
and. finally.
-411*
,!<//)-*> max [L±|]\
^#0 *-r[A)i
Corollary. If II Is a y-perfect hypcrgraph. then
«(//)-max [I'll]*.
a*a
Suppose A is a non-empty subset of X, Let q - r{A) -
i, /ta Av) is a strongs-colouring of //,,. then
- \AX | + - + | A, | < g»(//) - r(/l) a(W).

450 HVPIHCIRAI'HS
Hence,
a(//)>L^i M 0),
und
rl £ ||*
at(W) > max ~ .
jda L r(/l) J
From Theorem 1, it follows that
Q.E.D.
Remark, If hypcrgraph // is a graph, then the inequalities of Theorem I
are the best possible, since the expression
can attain the value of a(//) as well as the value of pill).
For example, consider the graph C, with 6 vertices shown in Fig. 20.1.
Fl||. 20.1
We have:
a(C,)-3 - MG|)
Let G3 be the graph obtained from G, by removing the isolated vertex. For
graph Ga, we have:
- 2 < fco(G2) - 3 -
2. Balanced hypergraphs
A hypcrgraph // is said to be balanced if every odd cycle
(fl,,£,,fl2, E3 £3,14.1. fli) has an edge £, that contains at least three
vertices a, of the cycle. For example, a family of intervals of points on a line

DAIANCII) HYPHUiRAI'HS AND UMMOOUI.AR IIYPIKORAPIIS 451
is u balanced hypcrgrnph. Clearly, a multigruph is balanced if. and only if,
the inultigraph is bipartite.
Proposition I. Let II be a balanced hypergraph, then erery partial hyper-
graph IV is balanced*
If //' had on odd cycle with no edge containing three of its vertices, then
this sequence would be also an odd cycle of // with no edge containing three
or its vertices. This contradicts that // is balanced.
Q.l-.D.
Proposition 2. Lei II he a balanced hvpergraph, then erery subhypergrapli
Hs is balanced.
If lls had un odd cycle with no edge containing three of its vertices, then
this sequence would define for // an odd cycle with no edge containing three
of its vertices.
Q.E.D.
Prnpnsition 3. Let If ■ (£i fie I) be a balanced hvpergraplu and let Fo be
a set. I'hcn the hypergrapli IV ■ (/:, u £0 f 16 I) is also balanced.
Suppose that //' has an odd cycle // with no edge containing three of its
vertices. Let this cycle be denoted by
At least one vertex x\ lies in £0 (otherwise, ft would define an unbalanced
cycle of //), Since k > 3. /« has an edge E', with./1 / 1 and / i, and £J
contains .vj, x',.i und x\. This contradicts the definition of/j.
Q.E.D.
Proposition 4. Let II ■ (£,, Fa £m) be a balanced hypergraph. and let
*o U £|. I'hin the hypergraph (£, u {.v0 }. £a £«) «" balanced.
The proof is immediate.
Proposition 5. Let II » (X, ,) be a balanced hypergraph. and let .v, X.
The hypergraph IV » (A". ') obtained from II by adding a new vertex x\
and by setting
A"- XKJ{x[}
E, if xt ? F,
£' \- £, u {*,'} if
also balanced.
The proof is immediate.

452 HYPERURAPIIS
Proposition 6. Let II m (xx, ..., xn; Ex £m) be a balanced hypergraph.
Then its dual H* » (fj <?«; A\ Xn) Is also a balanced hypergraph.
By definition,
X, -{*//< in. 4 3 x,).
Consider an odd cycle n - (ex, A',. ea ea,n, XaP»i. ex) in //•; it corre-
corresponds to an odd cycle (*,, £a, x3 £ap + i. Xap*i, £1, xt) in //.
Since // is balanced, some £, contains three of the x,> Therefore some ver-
vertex e, of ft belongs to three of the X,\ or, cquivulcntly, some XK. contains
three of the ?,. Thus, cycle ft is balanced.
Q.E.D,
Proposition 7. // //-(£,//€ /) Is a balanced hypergraph, and if H' m
(£( I is J) is a partial hypergraph such that ijej Implies Et C\ E, 0, then
PI £, 0-
(In other words, the edges of a balanced hypergraph satisfy I telly's property.)
The proof is an induction on the number of edges in //'. The proposition
is true for every subfamily with two edges. Assume that it is true for every sub-
subfamily with less than p edges; we shall prove it for a subfamily (E, / ieJ) <*
(E[,Ea E't) where/? > 3,
By the induction hypothesis, for every k < p, there exists a vertex ak with
PI £.'•
We may assume that the ak are all distinct (otherwise Pi E\ 0).
Consider the sequence
H - (a,, £J, a3, £;. «ra. £3. «i) •
If two of the sets £(. £J. £3 arc equal, for instance if £| - £J, then ox
belongs to OME't, and the proof follows.
If no two of the sets £J, £a, £3 arc equal, then /< is an odd cycle, and one
of these edges, say E[, contains {o,, aa, «a}, I lence,
fl|€ PI fy
Q.E.D.
Propositions 6 and 7 show that a balanced hypergraph is conformul.
Theorem 2. A hypergraph II » (X, >) w balanced if. and only if, for every
S^X. the subhypergraph //s fc blcolourahle.

IIAl.ANCliD HYPI KURAI'KS AND UNIMOUULAR HYPERGRAP1IS 453
If every subhypcrgraph of // is bicolourablc. then // is balanced be-
because otherwise, there exists an odd cycle (rt|.£|.«a £"„.«,) with no
E, containing three of the a,, and .5 » { o,, aa «p } generates a subhypcr-
subhypcrgraph //., with /.(Us) > 2, which is a contradiction.
Conversely, let // be a balanced hypergraph that is not bicolourablc, with
minimum order // » | X |. We shall show that this yields a contradiction.
1. We shall show that each vertex .v0 belongs to at least two dillcrcnt edges
of // with exactly two elements.
The subhypcrgraph //0 generated by A' - {,v0} is balanced (Proposition
2) and lms order« - 1. Therefore it has a bicolouring (S?. Sj). If .v0 did not
belong to an edge with exactly two elements, then (S? u {.v0}, S3) would he
a bicolouring of //, which contradicts the dclinition of //. If .v0 belongs to
only one edge with two elements, and if its other (Midpoint lies in 5?, then
(Si, S§ u { A'o}) would be a bicolouring of //. llcncc. .v0 belongs to at least
two edges with two elements, say [xo.y] and [.v0. r] with z y.
2. Denote by 0 the family of edges with exactly two elements. Consider
graph C » (A. //). Since G is balanced (Proposition I),C is a bipartite graph.
Consider a connected component C of G'o. Since G has order > 3 (from
part 1), there exists at least one vertex .v, in C which is not an articulation
point.
3. Consider the subhypcrgraph //, generated by X - {.v, }. It is balanced
and lias order // - 1. Therefore //, admits a bicolouring E,. Sa). and all
the vertices adjacent to .v, in G have the same colour. Let S, be the set of
vertices with this colour.
Then. (St. Sa \J {.v, }) is a bicolouring of // because each edge of // with
two elements is bicolourcd (since it is an edge of 0'). and each edge of //
with more than two elements is also bicolourcd (because its intersection with
A" - {.Vi} is bicolourcd).
This contradicts the assumption that // docs not have a bicolouring.
Q.E.D.
Theorem 3. For a balanced hypergraph II » (E, f i e /), let k » miii,a||£i |.
f'here exist k transversals of II that partition the vertex set X.
Let (.9|, S-2,.-, SJ be a partition of X into k classes, and let k(l) be the
number of classes which meet edge £,. If k(i) ■■ k for every /. then X is par-
partitioned into k transversals.
If k(!) < k for an index /» /0. then A(/o) < | /T,o |; hence, there exist
indices p and q such that:
I Sf n El01 > 2 .

454 HYPFRCIRAPIIS
and
I A", n Ele | - 0.
The subliypergraph //' generated by Sp u .V,, is balanced. Thus, by Theorem 2,
//' has a bicolouring (S'p, £,)).
Let S', » S, for 7 p.q. The partition (Si.-VJ SI) determines new
coclVicicnts k'(i) with
A'('o) - *(/„) + 1
*'</) > *(/) (/ #„).
By repeating this transrornution as many tiitics us needed, we obtain a new
partition (S[. S'a Si) with k'(i) - k for every /. Clciirly. this is the re-
required partition of X,
Q.E.D.
Theorem 4. A hypvrgraph II is balanced if. ami only if. y(U') - /(//")
for every partial suhhypcrgraph IV of II.
1. Let // be a hypcrgraph. and suppose that the nbovc equality holds Tor
every partiul subhypcrgruph //' of //. If hypcrgraph // is not balanced, then
there exists an unbalanced cycle, say.
/i » (fl|. £\. aa. £*a,.... a*. /:*, «|).
The partial subhypcrgruph //' > (S, >') defined by S > {«,, ai%.... ak] and
<$'■(£, n S. E3 n S Ek n S) is a graph consisting of an odd cycle.
Hence >(//') - 3. r(ll') » 2. which contradicts that y(//') - r(ll').
2. Let // be a balanced hypcrgraph of rank Ir, we have >'(//) > h since h
difTcrcnt colours arc needed to colour a maximum edge; therefore, it suffices
to show that there exists a strong /(-colouring of //.
Consider a hypcrgraph //' » (X\ >') obtained from // by adding a set
A, with | A, | » h — | £, | elements for every i and by letting:
A" - X\J U *i
i i
and
/-(Eiu/f,// /).
Clcnrly. //' is a uniform hypergraph of rank h. and by Proposition 4. //' is
balanced. By Theorem 3 applied to //'. there exists a partition {T{, Ti,.... Ti)
of X' into h transversals. Since | T', n E, \ < I for every / and every/ the sets
S, » 77 n X arc the classes of a strong /i-colouring of //.
Q.E.D.

BALANCED IIYPERORAPIIS AND UNIMODULAR HYPrRORAPHS 455
Corollary. If H is a balanced hypergraph with rank h. and ifhx + Aa — h,
then there exists a partition (Xx, Xa) of X such that
'(*,) - A, r(Xa) - /»a.
Let E,, Sa..... S,,) be a strong /i-colouring of //. Let
and
*■ - U $ ■
Clearly. (A^, Xa) is the required partition of X.
Q.F.D.
Application. Let G - (AC, ) be a bipartite mulligraph. and consider the
dual hypergraph G*.
Since G is balanced, hypergraph G* is also balanced (Proposition 6). Uy
applying Theorem 4 to 6'*, we obtain a well known result:
The chromatic index of a bipartite nudtigraph is equal to its maximum de-
degree (Theorem 2, Ch. 12).
Applying Theorem 3 to hypergraph 6'*, we obtain a result of Gupta
|I967]:
In a bipartite multigraph G with minimum degree A\ there exists a partition
of its edges into k classes, such that each class covers all the vertices of G.
Ihcorcm 5 (Rcrgc, I as Vcrgnas[l970]). Let II be a hypergraph. let v{ll)
denote the maximum cardinality of a matching and let x(ll) denote the mini-
minimum cardinality of a transversal. Then. H is balanced if, and only if.
for every partial suhhypergraph II' of II.
If the above equality holds, it is easily seen (hat // is balanced (the proof
is the same as for Theorem 4).
It remains to show that v(//) - t(//) for every balanced hypergraph //.
Consider a balanced hypergraph // - (/:, / ie I) with v(//) < r(//) in which
L«; I £i I is minimum. Let q ■ v(//). Thus, t(//) > q. We shall show (hat
this leads to a contradiction.
I. Clearly, £", $ I', for / j since £, c /••, implies that
Mil) - v(//. - 1:,) - x(ll - L,) - r(//).
which contradicts the definition of //.

456 IIYPERORAPHS
2. Also, | £', I > 2 for every iel.
If £, - {Xi), for instance, then from part I, x, £Ui>i £|. Therefore,
v(W) -
3. We shall show thai if So m (Etj jeJ) is a maximum matching of //
then it partitions the vertex set X of //.
Suppose (hat there exists a vertex a i (J.-»; E,, and let E', - £, - {a). The
subhypcrgraph //' - (£,'//€/) generated by X - {a} is balunccd (Proposi-
(Proposition 2). and from Part 2. it follows that
Thus, the matching rf"i - (Ej f J J) is not maximum in //', and by Theorem
(I, Ch. 18), there exists an odd maximal alternating sequence a' - (F[, £J,
l
Cl, Cg tn6 o
Fi, Fj,.... Fp + l e ' — • o •
Since the sequence corresponding to a' in // is not a maximal alternating
sequence by Theorem (I. Ch. 18), there exist indices Ar,, k2 < p + I with
Fkl r\ Fka - { a).
Let G be (he representative graph of the family of sets (£,, £a £"p,
Flt F3,.... /'„;), Graph G is connected and has an odd cycle. A minimal
odd cycle of G defines an odd cycle of // of the form:
(o. Ffl,xi. £/,,>■!, /',.,. x3. Elt /■*. a).
No edge of this cycle contains three vertices of the sequence, which contra-
contradicts that // is balanced.
4. Let .v, e £,, and consider the hypcrgraph
No cycle of Tl uses edge Ex, and therefore Jl is balanced. From Part 2
I \E,\< I \E,\-
i«; n/
By (he induction hypothesis, v(/7) - t(/7). Hence
Let o be a maximum matching of /7; then .,e o because »-(/7) > q ■ v(//).
Thus, o - { i} is a maximum matching of //. and the union of its edges
docs not contain a,, which contradicts part 3.
Q.F.D.

DALANCliD HYPrRClRAFUS AND UNIMODULAR HYPIRORAPHS 457
If // is a graph, Theorem 5 yields: In a bipartite graph, the maximum cardi-
cardinality of a matching is cquai to the minimum cardinality of a transversal
(Kttnig's theorem, Chapter 7).
Corollary 1. If II is a balanced hypcrgraph with a rank-function r(A),
then
Clearly,«(//) - v(//*) and /<(//) - t(//*). Since //* is balanced (Proposi-
(Proposition 6), Theorem 5 implies that i(ll) — p(H). From Theorem I. the required
equalities follow.
Q.E.D.
Corollary 2. Let II be a balanced hypergraph with m edges and with rank-
function r(A), and let k < m. A necessary and sufficient condition f>r the
existence of a wring with k edges is that
k r(A) - | A | > 0 (Ac X).
This follows from Corollary I.
Theorem 4 and Theorem 5 together yield a new result for graphs: Let
0* - (AC E) be a simple graph. Let' be the family of all its maximal cliques.
W A <= A'und J c f,, lot GAt/ denote the graph obtained from G by deleting
the vertices of X — A and the edges that do not belong to a clique in
Let a(G) denote the stability number of G (maximum cardinality of a stable
set), let 0(G) denote the partition number of G (smallest number of cliques
which cover A'), let y[G) denote the chromatic number of G, and let wF')
denote the density number of (I (maximum number of elements in a clique).
theorem 6. 77k* three following conditions are equival'nt:
1I) »(£„. ) ■ 0(GA. ) for every A and every B\
B) y(GAm ) - w(C/(. a) for every \ and every ;
C) every odd cycle in G contains at least one 'dge with the property that
•very maximal clique containing this edge contains a third vertex of
the cycle.
I ct // - {X. '6) be the hypcrgraph of the maximal cliques of G. Clearly.
C) is equivalent to
(.V) II is balanced
and to
C") the dual //* of II is balanced.

458 HYPrRORAPHS
Also, (I) is equivalent to
(O v(//') - t(//') for every partial subhypergraph IV of //*,
Clearly. B) is equivalent to
B') '/(//') - r(H') for eixry partial subhypcrgraph W of II,
By Theorem 4, B') is equivalent to C'). By Theorem 5, (I') is equivalent
to C"). Hence (I). B). and C) are equivalent.
Q.E.D.
Lovasz has studied the conditions under which each partial hypcrgrnph
//' of hypergraph // satisfies v(//') - t(//'). The class of hypergraphs with
this property is more general than the class of balanced hypergraphs.
The chromatic index q(H) is defined to be the smallest number or colours
needed for a colouring of the edges of //. i.e. q(ll) - )'(//*)• Let x denote
the family of edges that contain vertex .v. and let f>x(H) - | >x | denote the
number of edges that contain .v. Let
b\H) - max SX(H),
Clearly. q(H) > 6\H).
A hypergraph // is defined to be normal ifq(H') - H(H') for every partial
hypergraph //' of //. Fach partial hypergraph of a normal hypergraph is
normal, but a subhypcrgraph of a normal hypergraph need not be normal.
I vmma. //■//■(£,, £a, .... £n) is a normal hypergraph, anil if an edge
Eo - Ei is added to II, then the new hypergraph ll0 is also normal.
It suffices to show that if If is a partial hypergraph of // and if Hi ■
H' + Eo, then b\H'o) - q(H'o). We may assume that //' contains £, (other-
(otherwise, the equality follows immediately). Let
q -
Case I: Edge £! contains a vertex x with o'x(H') - q.
Clearly.
Hence,
«//£) < q(lli) < q(H ) + I - q + I - Wo) •
Thus,
/i) - <tUI'o) ■

DALANCCD IIYPERGRAPHS AND UNIMODULAK HYPKRORAPHS 459
CASt 2: ox(ll) < q - I for every x e £,.
Consider a (/-colouring or the edges or //' and suppose that edge £( is
coloured red. Each vertex a with £„(//') - q is incident to a red edge distinct
from Ei. If/\ denotes the set or red edges distinct from £,, then
<5(//' -*,)-*-! .
Since //' is normal, this implies that </(//' — <?t) — q — I. and there exists
a (q - l)-colouring of the edges of //' - ,. Since ^x u { £0} is a matching,
the edges of this matching can be assigned a cy-th colour; this produces a
^-colouring of //,',. Hence.
q -
Thus. *(/«)-?(//a. Q.E.D.
theorem 7 (Loviisz [1972]). M hypergraph II is normal If. ami only if,
»•(//•) - x(ll')
for each partial hypergraph IV of II.
Necessity. To show that each normal hypcrgraph satisfies this condition,
it suffices to show that If II is a normal hypergraph with »■(//) - q. then there
exists a transversal T with \T\ — q.
Since // is normal, its dual //* is y-pcrfect, and conformal (by Proposi-
Proposition. Section I), and consequently //satisfies the Hclly property (Theorem 3,
Ch. 17). Thus. i(//) - I implies that there exists a transversal T with
| 7*| - I, and the proposition is valid for q - I.
Let q > I. Assume that the proposition is valid for each normal hypcr-
hypcrgraph //] with v(//,) < q. We shall show that it is also valid for a normal
hypcrgraph // - (X, ) with v(//) - q.
If there exists a vertex x X with \{H - x) < q. then there exists a trans-
transversal Tof II - „ with q - I vertices, and 7*u {.v} is u transversal of //
with q vertices. Hence, the proposition is valid.
Therefore, we may assume that
v(//- <?„) - </ (.v X).
For each .v, e X, consider a maximum matching . , of // - X(. Let « -
I V|. Consider the hypergraph Ho - . t +• a +■•• + • * formed with all
the edges of the . , for /- 1.2 //. (Thus, hypcrgraph Ho contains
exactly nq edges.)
Since each vertex x, is not covered by • (. it follows that
o(H0) < n .

460 IIYPERORAPIIS
Moreover, in a colouring of the nq edges of //0, the same colour occurs
at most q times, and consequently, there are at least n different colours
Hence,
q(H0) >n> 6(H0).
However, from the lemma, //0 is a normal hypcrgraph, which implies that
<?(//o) ■ fi(Ho)- This contradiction achieves the proof.
Sufficiency. Consider a hypcrgraph // - (X,£) such that each partial
hypcrgraph //' of // satisfies v(//') - t(//'). It suffices to show that 6(H) m
q(H).
Consider a hypcrgraph R whose vertices correspond to the matchings in
H, and whose edge :, is the set of all the matchings in // th.it contain edge
£,. A subfamily //' - (E,f JeJ) has a non-empty intersection if. and only if,
Et^E,. 0 (j,f J)
(because this implies that x(H') - v(//") - I). This is equivalent to
E,rsEy-0 (i.feJ).
In other words, h sets of the family (E,} ieJ) have a common vertex in // if,
and only if. (E,f ieJ) is a matching in /7. Thus,
A) tf/7)-K//).
B) ft//) - v(/7).
Moreover, the sets of a family (EffjeJ) have a common vertex /0 ■(•
and only if, for each JeJ. edge E, belongs to matching >0, that is. if. and
only if. (FjfjeJ) is a matching in //. Thus.
C) </(//)- r(/7).
D) ,5(/7) - \{H).
By applying (I) and D) to a parti.il hypergraph //' of // and to the corre-
corresponding partial hypcrgraph IT of FI. it follows that
q( ')-&(!?').
Thus. /7 is a normal hypergraph, and from part I. \(FI) - t(FI). Therefore,
from B) and C). it follows that
Q.E.D.
Fournicr and Las Vcrgnas [1972] have also shown that a normal hypcr-
graph is bicolourablc. More precisely, if each odd cyclv of // contains three
edges with a non-empty intersection, then // i'.v Mcohurable.

BALANCED HYPCRORAPHS AND UNIMODULAR HYPERURAPHS 461
Theorem 7 has been used by Lovdsz to prove that a graph is a-perfect
if, and only if, it is y-perfect. More precisely:
Theorem 8 (Lovasz ([1972]). Let G - (X, E) be a graph such thai u(Gs) -
0(Gs) for each S e: X\ then y(G) - io{G).
Construct a hypergraph whose vertices are the cliques of G and whose
edge X{ is the set of all cliques of G that contain vertex x, £ X. Clearly,
graph G is a representative graph of the edges of //. Thus,
(I) v(//)-«(C),
B)
Furthermore, from its definition, hypergraph //satisfies the Mclly property.
Therefore,
C) <5(//) - w(G),
D) t(/Y) - IKG).
Fach subgraph Gs of G corresponds to a partial hypergraph //' of //, and
this correspondence is bijectivc. Applying (I) and D) to Gs and to //', it
follows that v(//') - r(//'). Therefore, from Theorem 8, // is normal.
Hence, q(H) - 8(H), and from B) and C), we have y(G) - w(G).
Q.E.D.
We shall now use the above properties of balanced hypcrgraphs to
determine a sufficient condition for a hypergraph to contain A- pairwi.se
disjoint transversals. This condition was first shown for k - 2 by Lovasz
[1968] and then extended to the general case independently by Lovasz [1970]
and Las Vcrgnas [1970]. First, a lemma is needed.
I cninm, let k > 2. let G - (X, Y, D be a simple bipartite graph such
that
| I'(S)\ >(k - \)\S\ + I (Sc X. S 0).
// no partial graph of G (other than G itself) satisfies this condition, then
each vert >x x In X has degree k.
l-'irst note that by letting S-{.v}. the above condition implies that
d,(x) > k for every vertex x.
Let .</ denote the family of all sets A <=■ \ such that
| l\A)\ -(A- - I) Ml + I .

462 HYPERCIRAPHS
I. Wcshallshow that if A, /4'e.e/and if A n A' 0, then A n A' e
We have:
u a1) | + | r(/i n /i') | < | r(A) u r(A') \ + | rM) n / (/O |
- (k - l)| A KJ A' I + I
+ (* - \)\Ar\A'\ + \.
Moreover, since G satisfies the condition of the lemma, and since A n A
0, we also have:
| r(AKJA')\ > (k - \)\AUA'\ + I
\ I\A n A')\ > (It - \)\ A n A" \ + \ .
Therefore, equality must hold in these two inequalities. Hence. A n /4'e.c/.
2. Consider a vertex a in X. and let /(«) - {ht,ba, ...,hp}. Let 6", ■
(X. Y, I',) denote the partial graph of G obtained by removing edge [a. b,]
There exists a subset A, of X such that
Hence,
At a a; A,et/\ h{ r{(A,).
Let A — /f, n i4a n ... n ,4P. From Part I. A e.t/. From above,
H/l - {«}) n /(a) - 0 .
Therefore.
I/(/I) | -</«(«> + M(-4 -{«})| >k + (k- \)\A -{a}\
Consequently, equality holds in (his inequality, and </«(«) - k.
Q.E.D.
Ilu-on-m 9 (Lovdsz [1970]. I as Vcrgnas [1970]). let k > 2, «/«/ /rf //
Eil is I) be a hypergraph such that
hypcrgraph // contains k pairwise disjoint trarersals.

BAI ANCED IIYPERtiRAPIIS AND UNIMODUI AK IIYPI KORAPHS 463
Consider a family //' - (£|//e /) such that
E/cE, (lei)
| U f,'| >(*- 1)| y | + 1 (Jcl,J 0),
nnd that is minimal with respect to the order relation //' < //" defined by:
E\ <= £" for all / 6 /. Consider the bipartite graph G,r - (/, X, I) where
/•(/) m £,'. Since this bipartite graph satisfies the conditions of the lemma.
I E't | - k for every / /.
Moreover, for each J <^ I, J 0.
2 (|E,'|- !)-<*- \)\J\ <l U Ei\~ I-
From Proposition E, C'h. 17. §2). this implies that hypcrgraph //' contains
no cycles. Therefore, //' is balanced, and. from Theorem 3, //' contains A'
puirwisc disjoint transversals. These sets arc also transversals of //.
Q.E.U.
3. Uninuxlular hypcrgraplis
This section treats a class of hypcrgraplis which, like balanced hypcrgraphs.
generalizes the concept of a bipartite graph. These hypergraphs which arc
important in integer linear programming give sharper results.
I'irst note that the concept of a (/-colouring of a graph can be extended to
hypcrgruphs in many diHerein ways.
A) A (f-fohuring of a hypcrgraph // ■(£",//€ /) is a partition (S,, S3
S\) of its vertices into q clusscs such that
/€/, /<</, \E,\ > I E^S,.
The smallest number of classes is the chromatic number x(H). See Chapter 19,
B) A strong q-vohuring is a partition E,. S3..... SJ into q classes such
that
|E,nS, | < I del. jKq).
The smallest number of classes is the strong chromatic number y(//). Sec
Section I.
C) An equitable q-coburing is a partition E,, S3.,.., Sv) into q classes
such that for each ie I and for each// < q,
- I < | £, n S, | - | £, n S,. | < I .
The smallest number q > 2 for which there exists an equitable G-colouring
is defined to be the equitable chromatic number k(ll) of hypcrgraph //.

464 1IYP1KOHAPHS
If // is a graph, then these three definitions coincide with the ordinary
definition of the chromatic number of a graph.
The reader can verify that the hypcrgraph // defined by the projective
plane with 7 points (Fig. 19.1) .satisfies
*(//)-3. y(//)-7, *(//)-7.
The reader can also verify that the hypcrgraph // with 9 edges in Fig. 20.2
satisfies
/(//)-2, }■(//)-4. A(//)-3.
Proposition I. For each hypergraph II,
k(H)
Clearly, each strong (/-colouring of // is also an equitable (/-colouring of
//. Similarly, each equitable (/-colouring of // is also a /-colouring of //.
The proposition follows.
Q.E.D.
Proposition 2. // there exists an equitable hlcolourlng of //, and if Gn
(/. X. /") denotes the bipartite graph of the vertex-edge incidences in II, then
for each partial hypergraph II' of //. the graph (!„- is either non-eulerian% or
euleriun with the nwnlwr of edges equal to a multiple of 4.
The bipartite graph G,, is defined by /'(/) ■ £",. A graph is said to be
eulerian if it has only even degrees.
If // lias an equitable hicolouring, then clearly //' also has an equitable
bicolouring. Let (S,. Sa) he such a bicolouring. For k ■ I, 2. let /■'„ denote
the set of edges of Gir that arc incident to a vertex .v Sk. If G,r is eulerian.
then each vertex / e / is incident to an equal number of edges of Fx und Ft,
Thus, | /•', | - | Fa I• Moreover.
l^i I- 2 4rW,
is even. Therefore, the number of edges in G,r is | Fx \ + \ Fa | ■ 2 | Ft |.
which is a multiple of 4.
Q.E.D.
If // is u graph, or if // is the dual of a graph, it is easy to show that // con-
contains an equiiahle bicolouring if, and only if, for each partial hypergraph //' »/
//, the graph Gu- is either non-etderian or eulerian with the number of edges
equal to a multiple of 4.

BALANCED HYPFRGRAPHS AND UNIMODULAR IIYPrHOHAPIIS
465
This result is similar to a theorem or Camion [I96S] for totally unimodular
matrices.
A hypcrgruph // - (X, ) is defined to be unlnwdular if for each S <= X,
the suhhypcrgraph II s admits an equitable bicolouring.
Example I. Clearly, a graph is unimodular if, and only if, the graph is bi-
bipartite. Thus, the concept of a unimodular hypergraph is an extension of
the concept of a bipartite graph.
Exampi.f 2. Consider a set X of points on a line, and a family fi -
(£,, £„ £'J of non-empty intervals whose union is A'.
The hypergraph // ■ (A', <f) is unimodular because for each S <=■ x an
equitable bicolouring of lls is obtained by alternately colouring red and blue
the points of 5 as they appear in order on the line.
Proposition 3. If H is a unimodular hypergraph. each partial subhypergraph
of H is also unimodular,
This follows from the definition of unimodularity.
Proposition 4. A unimodular hypergraph is balanced.
Suppose that hypergraph // is unimodular but not balanced. Then //
contains an odd cycle («,, Et, a3. E3 Ep, (h) of which no edge contains
three of the a,. Thus, the subhypergraph IIA generated by A - {a,, aa «„}
does not have an equitable bicolouring, which contradicts the unimodularity
oUI.
Q.E.D.
FIr. 20.2
The converse is not true. Consider the hypergraph // in Fig. 20.2. Each
odd cycle of H contains three vertices of edge £\. Hence, // is a balanced
hypergraph. However. // is not unimodular. l-lypcrgrnph // has a unique

466 IIVPEKOKAPIU
bicolouring of its vertices. (This can be seen by colouring vertex a red, then
successively colouring in red or blue vertices b. c, d, e,f g, h, i,j.) This bi-
colouring is not equitable for edge £,.
Theorem 10. If II is a hypergraph without odd cycles, then H is unimodular.
Since each subhypergraph or // contains no odd cycles, it suffices to show
that a hypergraph // - (A", ) without odd cycles has an equitable bicolour-
bicolouring.
There exists a function x,(t) such that £, can be written as
{x,( I), x,B) *,(/•,)}, where r, - | £, |.
Let. , be the set of pairs jc,( I )jc,B), x,C)x,D). etc.
Consider the graph G - (X, ). where -U i-
We shall show that graph G cannot have an odd cycle. Suppose that G
contains odd cycles. Let fi - [«,, aa «,) denote an odd cycle of G with
minimum length. Cycle n is elementary; suppose that it contains two dis-
disjoint edges in the same class ,. Without loss of generality, let [u,,ai+l]
and [a,. a,n]be these two edges. Transform , by replacing these two edges
by edges [a,,a(tl] uncl [a,,a,tl). Replace n by the sequence either
(tf,.tfa a,,a,,i at) or (u,.,,u,l3 a,,a,n) that has odd length.
Repeat this process as many times as possible. Upon termination, an odd
sequence is obtained that determines an odd cycle in hypergraph H, which is
a contradiction.
Since graph G contains no odd cycles, there is a bicolouring E,, Sj) of its
vertices. Clearly, (S,, Sa) is also an equitable bicolouring of //.
Q.E.D.
Corollary. A hypergraph H - (£, / iel) contains no odd cycles if, and only
if each hypergraph H' - (£,'/ / 6 /) with £,' c EJor all iel satisfies k(ll') < 2.
The condition is necessary because hypergraph //' contains no odd cycles.
The condition is sutTicient because an odd cycle of // (if it exists) would
induce a hypergraph //' that is a graph consisting of an odd cycle. Hence,
k(ll') - 3, which is a contradiction.
Q.E.D.
The following theorem and its corollary arc direct extensions to unimodular
hypergraphs of a property initially discovered by D. cle Werra [1970] for the
dual G* of a bipartite graph:
Theorem II. A unimodular hypergraph II has an equitable q-colouring for
each positive integer </ > 2.

BALANCED IIVPI KOKAI'MS AM) UNIMOMUI AK IIYPIIUiRAPHS 467
For q - 2, the theorem is clearly true. For q > 2. consider a uni-
modular hypcrgniph // ™ (/;, // /). and consider a partition (Sx, Sj S,)
of its vertices into / classes, l-or each./. A* < </, let
elk(l) - I S, n £, | - | Sh n £, |
c(/) - max ,.„<-,.„(/).
Clearly. <•{/) > 0. Iftd") for nil /e /. the puriition is an equitable (/-colouring
of hypcrgraph // iind vice versa.
Suppose that there exists tin index /0 with c(i0) > 2, and let r and x be two
distinct indices with e,,,(i0) - «(i0). Then, Tor all./.
| S, n £;01 < | S, n £@1 < | sf n £lD |.
Since // is unimodulur. the subhypcrgraph //' of // generated by the set
S, u S, has an equitable bicolouring (S't, S',). Let S', - S, for./ r, jr. The
partition (S[, S? S'q) defines as before coefficients e'lk(i). and fory r, j,
we have:
<\'/(/o) < fr,(/o) - I « <('o) - I
'/.('o) - I <; r(W - I
e'r,Uo) < I < *•(/») - I .
Since e'p,,,(i0) - ep.,(/0) for p. q r,s, the number of pairs (j,k) with
^mO'o) < <*('o) - I is greater thun the number of pairs (./', A') with e,k(ic) <
KW " I-
Moreover, for all / /0, ejv(/) < f(/). By repeating this transformation,
a partition with <-(/) < I for all / is finally obtained. This partition is an
equitable ^-colouring of //.
Q.E.U.
Corollary. //" //-(£,//6/) Is a imimothilar hypergraph, ami if
k - min,,; | f, |, then there exists a partition (Tx, Ta Tk) of the vrtex
set X of II into k transversals such that
Note that there always exists a partition of X into k transversals from
Theorem 3 (since a uniinodular hypergraph is balanced).
Moreover, from Theorem II. hypergraph // has an equitable ^-colouring
(Ti.Ta 7*0, and

468
Furthermore, since k - min | E, |, each T, is a transversal of //.
QE.D.
Application (dc Wcrra A970)). If G - (X, E) is a bipartite multigraph
and if q > I. then the edges of G can Ik partitioned in q partial graphs G,,
Ga G, such Iluit for each I < q and for each x e X,
r i i
: d0,
It .suffices to apply Theorem II to G*.
Let A m ((a1,)) be a matrix with m rows and n columns. Matrix A is defined
to be totally uninwiiular if the determinant of each square submatrix of A
equals + I, 0 or - I. If A is unimodular, then clearly each coefficient a\
of A must be + I. 0 or - I, since a\ is a minor of order I.
We shall state without proof two fundamental properties of unimodular
matrices.
rheorem of Hoffman and Krnskal [1956). An n x m matrix A is totally
unimodular if, and only if, for any integer m-rvctors h and h' and for any inte-
integer n-rectors a and a', each face of the polytope
{x I x e R"; a < x < a ; h < A x < ft'}
contains an integer point.
It follows that each vertex of this polyhedron has integer coordinates.
Theorem of Ghouila- Hour! [1962]. An n x in matrix A - ((o1,)) is totally
unimodular if, and only if, each set J c { |. 2, .,., n ) can he divided into two
disjoint setxJi andJ2 sucli that
I 2 a\ - 2 a\ | < 1 V < m) .
A consequence of this theorem is:
Properly 1. A hypergraph H is unimodular If, and only if, its Incidence
matrix ((a1,)) is totally unimmlulur.
Properly 2. If H is a unimodular hypergraph. then Its dual H* is also a
unimodular hypergraph.
This follows from Property 1 since the incidence matrix of //* is the
transpose of the incidence matrix of //.

BALANCED HVPERORAPIIS AND UNIMOUULAR IIYPIKGRAPHS
4, Stochastic functions
469
Let // - (X. ) be a hypcrgraph. A rcnl valued function f(x) defined on
X is said to be a stochastic function associated with H if
(I)
B)
0 </(*)< I (veX)
£ ./(*) - l (i - 1.2. ;...in).
»« f.i
The support of function /is defined to be the set
S>- {xlxeX.
Not every hypcrgraph has a stochastic function associated with it.
The problem of characterizing the supports S of // is due to J. Csinia
A970).
Example I. A square matrix
Ip\ p\
P - Up)))
P'\
Pi
Pi
l/3| p2 ... />, ... ft,/
is called hi-stochastic if the following conditions hold:
/ p) >0 (i < n.y < »)
31 P; -I (/ < »)
(I pj - I (' < «) •
lii-siochastic matrices have been used to generalize convex functions (cf.
Bcrge [1958]).
A matrix of order n defines a hypergraph // whose vertices are the cells of
the matrix and whose edges arc the sets of cells in the same row or in the
same column. Note that hypcrgraph // contains no odd cycles and is there-
therefore unimodular by Theorem 10. Clearly, each stochastic function of //
defines h bi-stochaslic mutrix of order n, and vice versa.
Lemma. // // - (X, tf) is a hypergraph that has a stochastic function, then
the support of this function contains a strongly stable transversal of H.

470 HYPfRORAPHS
Consider the incidence matrix of //:
*! ... X,
a-l:
m \ai
If // possesses a stochastic function/, then the vector p - (PuPa Pn\
where p, — f{xj\, satisfies
\a p-u.i i)
1@.0 0)< p*(l. 1 I).
Since matrix A is totally unimodular, the above system of equations has
an integer solution p'. The set
T-{xjlpj- I},
which has exactly one vertex in each edge is therefore a transversal ami is
strongly stable.
Q.F.D.
1 hcorcm 12. If fl is a unimodular hypergraph that has a stochastic function,
then each stochastic junction f(x) associated with H con be expressed in the
following form:
fix)
c-1
0 < p, < I ; £ p, - 1 .
i» i
where <p,(x) Is a @. 1 )-stochastlc function.
Clearly, the subhypcrgraph of // generated by the support S, contain* n
set 7", that is strongly stable and is a irnnsvcrsul (from the lemma). Let
</>,(.*) be its characteristic function, i.e.,
<P\{x)
I if x e T,
0 if x € X - T, .

BALANCED HVPEKOKAPHS AND UNIMODULAR IIYPERORAPHS 471
Let
Pi
The non-negative function f(x) - ptfdx) has more zero values than func-
function/(x). If this function is not identically zero, then it is proportional to a
stochastic function, and therefore its support contains a set T3 that is strongly
stable and is a transversal. Let p3 ■ min {/(.v) - pwfa) / x T%} > 0.
Similarly, define the function tp2(x), und consider the function
f(x) - px «»,(*) - Pi <pi(x) > 0 ,
etc. Finally, consider the function
J (x) - P\ <Pi(x) - Pi Vi(x) pk tpk{x)
Clearly,
I AW - X fix) - I Pj m | - I Pj.
x*C, xtt, J-] >
If fk(x) is not identically zero, then 1 - £/>/ > 0, and the function
1
is stochastic, Therefore, from the lemma, its support contains a set 7\4i
that is strongly stublc und is a transversal. With the characteristic func-
function </>„., i(a-), the procedure cun be continued as above, etc.
The procedure terminates only when the function./^(.v) is identically zero,
which implies thin
1 - I Pj- I Mx)-0.
Then, function/is equal to 2.*-1 AVu where 2 A ■ '•
Q.E.D.
Corollary. // // is a unimmtutar hypergraph, a set S Is the support of a
ttovhastic function if. ami only if. S equals the union of strongly stable trans-
rersals of //.
I. Let S be the support of a stochastic function of the form
- I Pi £
i-1 (-i

472 HYPERORAPHS
If we let T, m {x/(p,(x) m | }, then, clearly, S - U?-i Tu where each
set T, is a strongly stable transversal.
2. A set of the form S ■ Uf-i Ti is the support of a stochastic function
where <p,(.v) is the characteristic function of set 7*,.
Q.E.D.
Application I (G. BirkhofT. [1946]). Each bl-stochastic matrix P is a
barycenlre of permutation matrices Mt, i.e.,
P - I Pi M,
0 < p, < 1. I P, - I •
i-i
Consider the hypcrgruph H associated with matrix P as before (see Example
p. 469). The coefficients of P correspond to a stochastic function f of //.
Therefore, Theorem 12 can be applied to/, and the functions^*) correspond
to @. 1) matrices A/,, that arc permutation matrices. The equality follows.
Q.E.D.
Application 2 (Theorem of Perfect and Mirsky [1968]). Let M be a
@, 1) matrix of order n. There exists a bl-stochastic matrix P that has a zero
coefficient wherever M has a zero coefficient, if, anil only if, M is not of the
form
where A is a square matrix of order k, 0 < k < n, and where B is a non-null
matrix.
If matrix M can be fit into the above form by a permutation of its rows and
columns, then A/ is not u bi-stochastic mntrix ((/>')) because then the sum
of the coefficients p\ in A would equal k, and consequently the sum of the
coefficients p\ in B would equal 0. which contradicts that B is a non-null
mntrix.

BALANCED HVPBRORAPHS AND UNIMODULAR HYPERORAPHS
473
w -
20.3
Conversely, consider a matrix M ■ ((/»))) that cannot be lit into the above
form. Let // be the hypcrgruph whose vertices are the pairs (/. J) with m\ m \
and whose edges arc the sets of cells in the same row of M or in the same
column of M.
It suffices to show that for each vertex x - (/".,/') of //. there exists a
transversal Tx that is strongly stuble and that contains x. Clearly, this would
imply that X ■ U*«x Tx, and from Theorem 12. that X is the support of a
stochastic function.
Let TV - {i1 /11 < n}. and for each ieN - {;'}, let
W)- ://./</»../ j'.m',- I}c/V -{./'}.
This defines a bipurtitc graph (N -{/"}, N - { f}. /').
We shall first show that:
|r</)|>|/|
n-{r}).
Otherwise, there would exist a set / c fit - {/'} with | / | > | /'(/) |. and
there would exist a set J of columns with
J =/"(/) u {/} and |y|-|/|.
However, this is impossible since the matrix A obtained from M by re-
removing the rows of N - I and the columns of A' - J would not be of the
form allowed by the theorem.
Inequality (I) implies that the bipartite graph (N -{/'}. N - {J'}. /') has
a perfect matching (Theorem 5. Ch. 7). The set of vertices of hypcrgraph //
that correspond to the edges (/. j) of this mulching form with vertex .v ■
(/', /") a transversal Tx that is strongly stable.
Q.E.D.

474 IIVPCRORAPHS
EXERCISES
1. Consider a hypcrgraph H - (X.6) where X is a finite set of integer points in the plane
where I he sets of 6 arc of the form
or A*-{(.v.</)/<.y.</N*..ygN}.
Show how to construct an equitable bicolouring Tor hypcrgraph //
2. Show how to construct an cquitublc bicolouring Tor the dual of a bipuriiic multigruph.
3. Consider the hypcrgruph (X. rf) where the vertex set -V is the set of vertices of u graph
G, and where each edge is either u singleton ( x ) or a sci of vertices on a cycle of C (Unit
belongs to a cycle basis associated with a given forest A of G).
Show ihut // has an cquitublc bicolouring.
4. Let [X; £,. £» Fn) be n hypcrgruph such thai E, r\ E, Implies thai cither
E, c £,. or £/= Eu
l.ci(Y; f\, Fa...., /•'„) he a hypcrgr iph with the same property. Show that if .Vand Y
are disjoint, then the hypcrgruph [X u Y; /d v /•',. Et \J Fa,..., Flt v /•'„) is unlmodu-
lar.
5. Recall from Theorem A4. Ch. 3) that " if C is a simple connected graph, then Tor each
vertex .v0. there exists a spnnning tree 7"nnd a direction of the edges of C such thut:
A) ris an urborc«ccncc with root xa. and
B) each cycle of the cycle basis nssociutcd with 7* is a circuit."
Let C be n connected gruph and let Tbe a spanning tree of C. Let A', denote the .set of
edges on a cycle ^ belonging to the cycle basin ussociuicd with 7'. Show thsil 7"can be
chosen so that (A'i".£'„....) Is a unimodulur hypcrgraph.
6. Let //-(*. ) be n hypcrgraph whose vertex set X is the set of Integer points (x. y) in
the plane with
0<.r</>. 0 < y < tj
and whose edges arc rccunglcs of the form:
Eia.a.b.b) ~{(x.y)j a < x < «' ;b < y < />'}.
Show that there exists an equitnble bicolouring for hypcrgraph //. (In general. // is not
unimodular.)
7. Show that a y-pcrfect hypcrgruph. uniform or rank h. has tin cquiinhlc bicolouring.
(If the hypcrgraph Is not uniform, then it might not have an equitable bicolouring. Sec
Fig. 20.2.)
8. Let G - {X. E) be a simple connected gniph. mil let illx.y) denote the dismncc
between vertex .v anil vertex .v (sec Ch. 4. $4). The chromatic number y'iG) of tlhiance p is
defined lo be the smallest number of colours needed to colour the vertices such that no
two vertices x and y with I < <Hx. y) < p have the same colour.
Show that y"(C) is the strong chromatic number of some hypcrgruph. Also show thnt
y"(G') - p + I if. and only if. one of the following conditions is sutisticd:
(I) | X\ -/>+ I.

BAI ANCI'D HYPKRCIRAPH3 AND UNIMODULAR IIVPERORAPHS 475
B) C is an elementary chain or length > p.
0) C is un elementary cycle whose length is a multiple of p + I.
(F. Kramer. H. Kramer A9691)
9. Let H m (X. ) be a unimodular hypcrgrnph. and lei p, (j ■ 1. 2, ..., n), b,, e, (/c /)
be non-ncgntlvc integers with b, < fl. For K c /, let:
KM -{llleK.E,3x)
*k.k- - { x I x e AT. I Kix) I > | KM I}.
A. Hoffman has shown thai:
A ncceunry and sufficient condition thai each vertex x can be nssigncd an integer
/K.v) with
0 *p{.x,) <p, (J 1,2 «).
hi < p(fi) < fi (/e/).
is that MA') - r<A") < p(^k. k<) Tor all A. A' c / such ihut
An A - 0 ,
II A(v)l —i A(.v)|| < I tv«AT).
Show that this coniliiion is necessary even if hypcrgruph // is not uniinodulur.
10. Use Theorem 3 to prove the following result:
Let X be a set with cardinality An, and let (/fi, /fa A,) and (/fn(.i.^n.9 ^j,)
be two partitions of X with \A,\-k for all /. Then, there exists a sol 7* such ihut
I 7' n A, | - l for all I.
(B. I., vnn der Wncrden. ,,l:.in Saiz ()l>er KUwscncinicilung von endlichen Mcngcn,"
Ahh. muih. Stm. Un!\>. Hamburg. 5. IV27.)

CHAPTER 21
Matroids
1. Matroid on a set
To study axiomatically the properties of linear independence, Whitney
[1935] introduced the concept of a mutroid. Let £-{<?,. e3 e„ } be a
finite set and let be a set of subsets of £, Set is defined to be a matroid on
E if the following conditions hold:
A) {e,}eP (/- 1.2 m)
B) FeP, F' 0, F' cF »» Fef
C) For each 5 <= £, the members of that arc maximal in 5 have the
same cardinality.
The pair M - (£. ■ ) is called a matroid {on set E), Clearly, a matroid on £
is a hypcrgraph/1' In particular, the rank function r(S) of a matroid is defined
to be
r(S) - mux | F n S | ,
Ft*
Axiom C) implies that each member of the family. that is contained in 5
and maximal in S has cardinality r(S).
The elements of £arc called the elements of matroid A/, nnd the members of
arc called the independent sets of matroid M. The independent sets corre-
correspond to the edges of hypcrgraph (£. ). The subsets of £ tlmt do not belong
to & arc called dependent sets. A minimal dependent set is called a circuit.
Proposition 1. // A/ - (£.. ) is a matroid with rank r(£). the maximal
Independent sets of M form a uniform hypergraph of rank r(£),
The proof follows immediately from the definitions.
(l) If axiom A) is eliminated, and if 0«. , I hen Is said to be a monoid In E (or a
pregeometry in the sense of Crapo, Roiii), In this case, I he matroid is no longer a hyper-
graph. Later on, n will be evident that Axiom (I) is the least important. With Axiom (I).
. is somewhat more than pregeometry but Is not yet a combinatorial geometry in the sense
of Crapo, Rota. A combinatorial geometry would require that each subset of two elements
or £ also belong to
In this chapter, we shall confine ourselves to treating only the combinatorial aspects of
niatrolih considered as hypergmphs. The nxiom system used here is due to J. Edmonds.
476

MATROIDS 477
Proposition 2. If M — (£, ) is a matroid with rank function r(S), then the
suhhypergraph A - {Fr\ A / Fe ,Fr\A 0} of M generated by
A <z Elsa matroid of rank rA(S) - r(S).
The proof is immediate.
Proposition 3. If M — (£. ) is a matroid of rank r(S). then the k-section
lk)-{FI\*\F\*k. Fe }
Is a matroid of rank r«w,(S) - min { k, r(S)}.
The prooris immediate.
Example I. The family '(£) of all the non-empty subsets of a set E is a
matroid of rank rE) - j S |, and its strong stability number equals I,
The family . <*,(£) of subsets of £ with cardinality < k and > I is also a
matroid because it is the ft-scclion of the preceding matroid. Its strong
stability number equals I. Each circuit is a subset of £ with k + I elements.
Exampi n 2. Let E be a finite set of vectors 0. Let • be the family of sets
of linearly independent vectors. Then, (£.',. ) is a matroid and the rank r(S)
of a set 5 <= e is the dimension of the linear space generated by 5. The strong
stability number of (£, ) is the maximum number of vectors of £ that are
in the same 1-dimensional subs pace.
Exampu 3. Let G - (X, E) be a multigraph. Let be the family of sets of
edges that do not form a cycle. Then. (£. ) is a matroid with rank r(S) equal
to the cocyclomatic number of the partial graph generated by S. Each inde-
independent set of this matroid is a forest of G. Each circuit of this matroid is an
elementary cycle of G.
Example 4. Let G - (X, E) be a multigraph without isthmi. Let. be the
family of sets ofcdgcs whose removal from G docs not increase the number of
connected components. Then, (£, ) is a matroid with rank r(S) equal to the
cyclomatic number of the partial graph generated by S. Each independent set
of this matroid is a coforest of G. Each circuit of this matroid is an elementary
cocyclc of G.
Fxampik 5 (Edmonds. Fulkcrson [1965]). Let G - {X. U) be a graph with-
without isolated vertices. For each mulching V c V. let 5A ) denote the set of
■saturated vertices in matching V. We shall show that the sets of vertices of G
that arc contained in S{ V) for some matching Fform a matroid M - (X.. ).
Let A <= X. Consider two sets F, and Ft of. that arc contained in A and
arc maximal in A. We shall show that | F, \ - | Fa |. Suppose the converse
occurs, for example, \ F3\ > \ Fx |. Then
I Ft - /-, | > I /■', - F, |.

478 IIYPFHC1HAPIIS
There exist two mulchings K, and Va such that
f, - AnSiVt). F, - AnS[V,).
At least one connected component of the partial graph generated by
is an open elementary chain n - n[a, b] with an endpoint a in Fa - F, and
the other endpoint b in F{ - Fa, from the lemma to Theorem (I.Ch. 7).
Therefore.
V\ -iVt -/i)u(^n;i)
is a matching that saturates all the vertices in F{ u { a). This contradicts the
maximality of Fu and completes the proof.
Exami'I !• 6. Consider a family {A, fj e Q) of subsets of a set E such that:
-4@ - \J A,-E.
A subset T - {tltta,.... tk) of f is called a /wf/a/ transversal if there exists
an injection7A) from { I. 2..... Ar} into Q - { I. 2 </} such that
f,€/f,(() (/-I. 2 A).
We shall show that the family of partial transversals is a matroid on E with
rank
|0| + inin (\AiJ) n S\ - \J\).
This matroid is called the transrersul matroid of the family {A, / / Q).
Consider the bipartite graph {Q, E. O where F is defined by:
/(./•) - At (ye Q).
From Example S, we know that the sets of vertices that arc contained in
S{ V) for some matching V define a matroid. The family of partial transversals
is a subhypcrgraph ofthis matroid. Therefore, the family of partial transversals
is also a matroid. From the Konig theorem (Ch. 7. § 3). the rank-function of
this matroid is:
r{S) - min (| Q - J \ + \ F{J)n S \) - q + min (| A(J) nS\-\J\)-
JCQ JCQ
Examplu 7. If (C>. Ca Cr) is a partition of a set f into p classes, and if
c,. t'a,.... cr arc integers with 0 < <•, < ' C, |. then the family
. -{F//c£. F 0.\FnC,\*ct for all/}
is a matroid on Ewith rank
>T min{f,.|SnC,|}.

MATKO1M 479
Clearly, if a set F . is contained in S and is maximal in S, then
| F n C, | - min { c,. I S n C, I},
and its cardinality is
I/-I-Z min{f,.|SnC,|}.
/-i
This completes the proof.
The following two propositions will he needed later.
Proposition 4. If M m (F,. ) is a matroid, its rank function r(A) satisfies
the following properties:
B) r({*})- I.
C) A c B -> r(A) < r{B).
D) r(A) + r(B) > r(/« u B) + r(/f n fl).
F^opcrtics A). B) and C) arc evident. To demonstrate Property D). let /
be an independent set contained in A <~\ B with \ F\ - r{A n B). Let FA be
an independent set that contains Fund is contained in A, with cardinality
IF,!- r(/f). l-ct £0 be an independent set th.it contains FA and is contained
in /(Ufl, with cardinality | £0 I - r(/l u A).
Clearly, Eor\ A - /■',, (since FA is a maximal independent set in A).
Clearly, En n {A r\ B) — F (since Fis a maximal independent set in A <~\ B).
Thus,
r(A u B) - | Eo | - | (£0 n /I) u (£'o n B) \ -
- | Lo n A | + | I-o n fl | - | Fo n /f n fl | < | FA \ + r(B) - \ F\ -
- r{A) + r(B) - r[A n fl).
Hence, Property D) follows.
Q.E.D.
It can also be shown that properties (I), B), C), D) characterize the rank
function and could be used as axioms to define a inatroid on £.
Proposition 5. Chen a inatrnhl M - (£,. ). if F . o/k/ if FU {« } £. .
//if/; F u { o } contains exactly one circuit.
Let Fbe a minimum independent set that contradicts the proposition. Then,
Fu{a) contains two distinct circuits Ci and Ca. and therefore aeCi,
a e Ca. From the minimality of C, and Ca, there exists an clement a{ e Ct -
Ca und an clement oa 6 Ca - Ct.
I. We shall show that the set /l0 - Fu {«}-{«,, oa J is independent.
Otherwise, consider the set F' - F - {a{), that is independent since it is

480 HYPEHOKAPHS
contained in F. The set F' u { a} contains circuit Ca and a minimal dependent
set of/fo> Therefore, F' contains two distinct circuits, and since | F' | < | F |,
this contradicts the minimality of F.
2. The suhmatroid generated hy /-'u {a} has rank | /■' | and contains the
independent set Ao. Since | Ao | < | F\, it follows that, for / - 1, 2,
This is impossible since C, is a dependent contained in Ao u { a,}.
Q.E.D.
Lemma. If S is a maximal strongly stable set of mutrohi M - (£, ), then
se S is adjacent to all a e E - S.
Let
5 — {s{, s2,..., sf}
be a maximal strongly stable set.
Consider a vertex a E - S. Since 5 \J { a) is not strongly stable, there
exists a vertex s, e S that is adjacent to a. For k y* j, vertex sk is adjacent to
{a,s,} bceausc the set A - {a.st, sk} has rank 2, and an independent set
that contains sk is contained in a maximal independent set F such that
| Fn A | - 2. Therefore, a is adjacent to sk for nil k.
Q.E.D.
Theorem 1. // A/ - (E, ) to a matroid with strong stability number
a(M) > i | £ |. '//en a(.W) - p{M), covering number of M.
Consider a maximum strongly stable set
5 — { st, *j,.... sp } .
Let
E- S- {a,, a, a,}.
Thus, </ < p. From the lemma, there exists an edge Ftl that contains a, and
j;. Since E can be covered with the p edges f,,!, Fa.a. •••. Fv.v, F,.,*it >-.
F,iP, then
/>(M) < p - a(M).
Since/>(//) > a(//) for each hypcrgraph //. it follows that p{M) - a(A/).
Q.E.D.
Theorem 2. >4 matroid M « (E, ) i* confornial (/", aw/ «///>• ^", rAfrf fxists
a partition (SuSa, ..., Sq) of E such that is identical to the family of non-
nonempty subsets F of E such that
\FnS,\ < I (/- 1.2 q).

MATKO1DS 481
I ct S, - {»i, sa,.... sf} be a maximal strongly stable set of a conformal
matroid M with rank h.
It suffices to show that family has the form required by the theorem.
1. Let F, be a maximal independent set that contains vertex j,. Let
A - £ - S,
A, - F, n A ,
Thus, | F, | - *. and | /!, | - // - 1.
2. We shall show that A{ is a maximal independent set in A. Otherwise,
there would exist an a e A with
A | u [ a } e & .
By the lemma, vertices a and sx arc adjacent and, therefore, arc contained
in a maximum independent set F,,.,,. Since matroid A/ is conformal, it
follows from Theorem B, Ch. 17) that there exists an Fo e. such that
FQ a [F nM,u{«})]u[M|U{ « }) n F.ilt] u (F,.,, n F) -
- Ax u { a ) u {s| }.
Hence, | Fo | > h + I, which contradicts that matroid M has rank //.
3. Since r(A) - A - I, each maximal independent set Fsatisfies
|Fn5,|- I.
Consider maximal strongly stable set S2 in the submatroid generated by A
(that has rank h - I). Using a similar argument as above, it follows that
|FnS,|- I
In this way, a partition (S{, S3 5h) of E can be defined. Clearly, each
maximal independent set Fof M satisfies:
\FnS,\- I (/- 1.2 //).
Hence, E1( 52 Sh) is the required partition.
4. Conversely, each set Fthat satisfies the above equalities consist of pair-
wise adjacent vertices. Since matroid M is conformal and has rank k, set F
is a maximal independent set.
Thus family. has the form required by the theorem.
Q.E.D.
2. The Kado theorem and related results
Let s/ - {Al% Aa,.... Aq) - (A, / /e Q) be a family of subsets of a finite
set E. A family of distinct representatives of. is defined to be a family

482 IIYFCKCjRAPIIS
(</(/) / / (?) of elements of E such that
B) aii)eA, (/- 1.2 q).
The clement a{i) is called the representative of set A,. Clearly, a set of
distinct representatives is a partial transversal of cardinality q. but the con-
converse is not true.
Consider the bipartite graph ((?, E. F) with e F(i) if, and only if, ee A,.
A set of distinct representatives is the image of a matching of Q into £(see
Ch. 7. §3). For J <= Q, let A(J) - U Ar Rcca" from Theorem E. Ch. 7).
that a necessary and sufficient condition for the existence of a fainily of dis-
distinct representatives is that
\A(J)\>\J\ iJczQ).
The following theorems generalize this result.
theorem 3 (Perfect [1969]). Let M - (£. ) he a matraid of rank r(E).
Let k he an integer < r(F), and let . ■ (AX,A* A,) - (At/I Q)
he a family of q subsets of E. A necessary and sufficient condition for M to
contain an independent set F m { a(i) / i e K}, A <= Q, | k | -ft, such that
a(i) 6 A, for all i e A. is that
r{A{J)) >\J\+k -q (J c (?).
Necessity. If there exists such an independent set F, then
r(A(J)) > | /■" n A{J) \ >|An./| — |A| + |^| — \ Kr\J\ >
Sufficiency. Suppose that the condition of the theorem is satisfied. Con-
Consider the families. - (BJieQ) with
B, c A, (/ e Q)
r{BU)) >\J\ + k-q (J c Q) .
Let & < ' signify that B, c B[ for all i e Q. Clearly, the relation ■< is an
ordering. Consider a fainily ■(£,,£, /}„) that is minimal with respect
to this ordering. We shall first show that | B, \ - I for all /.
Suppose that \ Bt\ > I. Then, there would exist two distinct vertices /;' and
b" in fl,. Let
B\ - B, - { h' }
B'i - Bt - { b- }
B\ m B]' m B, if / »* I .

MATROIDS 483
Prom the minimality of. , there necessarily exist two subsets /. J of Q such
r(BV))<\l\ + k-q,
r{B"(J)) <\J\+k-q.
Hence,
r(B\l)) + r[B~iJ)) < | /1 + | J \ + 2(k - q) - 2 .
Moreover.
B(l)v B"(J)m D(lvJ).
BV)nB'(J) => B{InJ - { 1 }).
Combining these results with Proposition 4 yields
r{BV)) + r(B'(J)) > r(B'(/)u B"{J)) + r{B'(l) n B"[J))
> r(B(l u J)) + r{lHl n J - { I }))
> I / u J | + | / n J - { I } | + 2{k - (i)
*\l\ + \J\ + 2(k~q)-\,
which is a contradiction.
Thus, we have shown that .-it is of the form ({b,}/1 Q). If we let
Bm{b,/leQ).
then from (I) it follows that
r(B) - r(B(Q)) >\Q\ + k-q-k.
Therefore, there exists a subset A of Q with | A' | -A- and an independent set
F -{/;,// A'} contained in B, such that:
b,eA, UeK). Q.E.D.
The well-known Rado theorem follows immediately from this result:
Theorem 4 (Rndo [1942]). If M - (£, ) is a matroid. then a family
. ■ (/f|, A a A,,) of subsets of E hatv an independent set of distinci
representatires if, and only if,
r(A{J)) > |./ | (;c0.
The proof follows by letting k - q in Theorem 3. Q.E.I)
Corollary 1. Two families •«/ - (Ax, A.t Av) and- - (Bt, Ba Bv)
hare a common set of distinct representatives if. and only if,
| A(J) o ft A') | > | J | + | A | - q [J,Kc Q).
Consider the transversal mntroid M of family . (see Example 6). The rank
of this matroid is
r(S) - q + min (| B(K) o S | - | K \).

484 HVPCKCJKAPHS
From Theorem 4. there exists a transversal set oft-/ that is independent in
M if, and only if, for all J c Qh
r(A(J)) - q + inin (| A(J) n H[K) \-\K\)>\J\.
Q.E.D.
Corollary 2. //" *• (Ct,Ca Cp) is a partition >f a set E. and if
ct, ca cp arc integers with 0 < r, < | C, | /w «// /", f/i«f a family. m
(At, A3 Aq) has a set Tof distinct representatives with \ Tr\ C, | < c, for
all /, //, ami only if,
f min { c,,M(./)nC, |} > I J| U c Q).
Consider the mairoid .1/ formed from the sets F c ^ with | Fn C, | < c,
for all / (see example 7). whose rank is
r(S)~ £ min{f,,|SnC,|}.
I-1
There exists a set of distinct representatives of. that is independent in 1/
if, and only if, for all J c Q%
r(A(J)) - £ mm {<-,. I A(J) nC,\)>\J\.
i' i
q.e.d.
The Corollary to Theorem F, Ch. 7). can be restated in terms of sets of
distinct representatives as follows: A necessary and sufficient condition fora
set B c £ to he contained in some set of distinct representatives of a fanuly
.of m (Au At Av)is that
min { | A(J) u B \ , a - \ B - A(J) \}>\J\ U c Q).
We shall extend this result to nutroids. hut first a Icinnu is needed.
Lemma. Let M - (£, ) he a matroid of rank r. let B . , and let
a > | B \. Then, the family
&B.<-{FIF<zE,Fvlie. . | F u H | < q }
is a matroid on E with rank
rHill(S) - min { r(S u B), q } - | B - S |.
Let S c £, and let 50 be a subset of 5 that belongs to „.„• If Fsatisfies the
two following conditions:
A) Fe M
B) SocFcS

MATROIDS
485
then, clearly F satisfies
|F| < min{r(S\J B).q}-\B-S\.
Thus, it remains to show that equality holds for some F.
Fill. 21.1
Set B u So is independent in M and. consequently, is contained in an inde-
independent set F' of B u S with cardinality
| F• | - r(S u B).
Let F" be an independent set such that
8uScc r c F , |F" |- min{r(S\jfl).</}.
Clearly, set f - F" r\ S satisfies conditions (I) and B), und lls cardinality is
|T| - min{ r(S u B),q} - | B - S \ .
Q.E.D.
Theorem 5 (Las Vcrgnas [1969]). Let M - (£,. ) be a mutrokl with rank
r. Let Be . ami let q > | B |. Then, a family .<• - (/f,, A2 Av) of sub-
subsets of E Afljv a set T of distinct representatives such that Te. and T = B if
ami only if,
min{r(/f(J)vjB).?} - \A(J)- B\ > I J\ [J <= Q) •
Consider the family
. ,,-{f/Fc£,fuBe. .IFvjUK*).
By the lemma, this family is a matroid on £.
If there exists a set 7" of distinct representutives ofV with Te and T = B,
then | 7~ | ■ q. Therefore,
Conversely, if there exists a set T of distinct representatives of .</ without
Te „,„ then
eP, \TvB\*q, \T\mq,

486 IIVPFRCRAPIIS
iind. consequently.
76 . T^B.
Then, from the Rado theorem, there exists a set T that satisfies the above
requirements if, and only if. the rank rBmV of the matroid (E,. OmV) .satisfies
>•„„( UJ)) > | J | (J c (?)
or
min { r(Atf) u B), q} - | B .1G) | > \ J \.
Q.F.I).
3. Image of a malroid
Many of the ideas to be presented in this section originated in the work of
l-dmonds and can be found in F.dmonds. [1965], [1970].
Let A/ - (E, ) be a matroid on E - { ?,, et em}, and let tp be »
mapping of E onto n set F.. Consider the family
^-(,/,(/■)/Fe. ).
Since tp is a mapping onto E, the pair ^»(i, ^) is clearly a hypcrgrnph,
called the image of St.
Ihcorcin 6 (Nash-Williams [1966]). If S7 - ( ;.,^) is the image of a
matroid M - (A",. ) hy a mapping <p of E onto E. then Kf is a matroid. and its
rank is:
HE) m min {tit? ' (A)) + I £ - A~ |).
I. We shall show that
max | F | - min (r(p~ '(/I)) + | J. - ~A
Clearly, max | F\ is the greatest integer Ar such that the family
has a partial set of distinct representatives that is independent in M and has
cardinality k. From Theorem 3, this is the greatest integer k such that
min (/■{</>-'(I)) + | £ - 11) > k.
Hence,
max | F | - min (/-(<p" !U)) + | £ - A I).
2. To complete the proof, it suffices to show that the image of M by tp is a
malroid.

MATRCIDS 487
Consider a mapping <p of £ - {elt e en} onto E - { e3, ea em)
such lhal
f>(ei) ■ e, for / 1 .
Mapping <p is said to be an elementary contraction. Since each mapping is a
product of elementary contractions, it suffices to show that the image of
matroid A/ by an elementary contraction ip is a malroid.
Consider an independent set Fo e such that Fo is maximal in. .We shall
show that ro is a maximum set. From part I, it suffices to show that there
exists a set 7 c £ such that
\e-a\.
In order to simplify, let £0 - E - {<•,, ea}. Since Fo is maximal in .iP, we
may assume that Fo is a inuxiinum set of. .
We may also assume that Fo contains both f, and f2 because, otherwise,
Fo would be maximum, since
I ?0 | - | Fo | - r(£) - rfo-'<£)) + | £ - £0 | .
Three cases must be considered:
Casl 1. r(£0) - r(f). Since
it follows thul r(/f — {<•,}) « r(£). Since Fo contains both (^ and c3, then
Consequently, there exists a maximum independent set Fo that docs not
contain *', and that satisfies:
Thus. Fo m P^ and
I ^o I - I n I - I n I - r(f) - r(^-'(£)) + | £ - £0 | .
Hence, the required equality is proved.
Cask 2. r(£0) * r(E) — I. Then, each maximum independent set contains
cither <fi or ea. Moreover,
! f'o n Eo | - | Fo | - 2 - r(/O - 2 < r(£0).
Thus, there exists a vertex tie Eo such that (Fo n £0) u {o} is an inde-
independent set with cardinality r(£0) - /■(£) - I. Let Fo be a maximum inde-
independent set that contains this set. Since F£ is maximum, it contains c, or c3.

488 HYPERORAPHS
Without loss of generality, we may assume that
Clearly. F£ = Fo. Therefore. Fo ■ F<5. and
\F'<y I - I Fo | - r(£) - ^"'(E)) + | £ - £ | .
Hence, the required equality is proved.
Case 3. r(E0) « r(£) — 2. Then, each maximum independent set contains
both <•, and e3. Consequently, Fo = £ and
Hence, in all cases, f0 is a maximum set of .'f.
Q.E.D.
If //' - (X, l). //■ - (A\ a) //" - (*. rf"') arc hypcrgraphs on a
set X, their join is defined lo be the hypcrgraph
U m //i v Ha v-v //' - (X, V «f).
i-i
where:
Clearly, H is a hypcrgraph on X.
Theorem 7. //"(£,. '). (£, a) (£.. ") are matroids whose ranks are
respectively rl, r3 rp, then their join hypergraph is a matroid, audits rank is
r(E) - min (r](A) + - + r"(A) + \E - A \).
Make p identical copies £', F3 £" of set £. Let 4 denote the element
of £' that corresponds lo eh e £. Consider the mapping <p of Uf. t £' into £
that maps pj,e £' onto fk. Clearly, M - (|J £', V ') is a matroid, and its
rank is
^U Z) - r\El) + r2(£2) + - + r'(E>).
From Theorem 6, the image of this malroid by <p is a matroid H, and Ki is
clearly the join Vf-1 • '■ Moreover, the rank of the joint matroid M is
'HE) - min(/-(v>"'(/l)) + I E - A |)
mini V /'(/I) + IE - >!|).
Q.E.D.

MATROIDS 489
Corollary I (Edmonds [1968], Nash-Williams [1968]). For a matroid
M - (£, ), the covering number p{M), i.e. the minimum number of indepen-
independent sets that are needed to cover E is equal to:
max f!-~l| .
By definition, (>(.\f) is the smallest integer k such that the join matroid
M V M V "V A/ of A; identical copies of matroid .W has rank | £ |. From
Theorem 7, (>(\f) equals Ihc smallest value of an integer k such that
min (hiA) + | E - A |) - | E | ,
ACT.
or. cqiiivalcntly, such that
m\n (kr(A\ - \A |) - 0.
A*K
This is equivalent to:
kr(A) - M | > 0 {Ac E).
or
fc > j»ll M c £. M 0).
Thus.
Q.E.D.
Corollary 2. For a matroid M ■ (£, ). f//r maximum number Ao of pair-
wise disjoint maximal independent sets is
■ r If" ^M
■ mm -'^- -V, .
4ck lr(£)-r(/4)J
Clearly. /c0 equals the greatest integer k such that the joint matroid of k
identical copies of malroid M has rank kr(E), where
min {kr(A) + I t - A |) - itrtf).
This is equivalent to
min (M^I) - ME) + | E - A |) - 0.
A<=r.
or to
k(«E) - r(A)) < | E - A | Mc£).
This yields the required equality. O E D

490 HYPCRORAPIU
Corollary 3. Consider a matroid M - (£, ) and a sequence *,, k2,.... kq
of integers such that
r(E) > fc, > fc2 > "• > fc, > 0. £ k, - | £ | .
Let kf denote the number ofk, that are > j. Then, set E can be partitioned into
q independent sets F,, F2 F,, with \ F, | - kt for all i, if, and only if,
£ fc? < | £ - A | (Ac E).
Consider the A,-scction A/,k|) composed of the family
It is a matroid. and this matroid has rank r\A) - min {r(A), k,}. There exist
q sets that satisfy the conditions of the corollary if. and only if, the join
matroid M «- VT-i M&o nas rank | H |. This is equivalent to
min(£ r'(A) + \E- A\)-\F.\,
which is equivalent to
£ min (r(/l). k,} + \ E - A \ > \ £ | - £ k, - £ k] (A c £)
i-l (-1 >>0
or to
£/7 - '£/? < I £ - a i uco.
Q.E.D.
The corollary follows.
Corollary 4. Let //M be a hypergraph c msisting of the circuits of a matroid
M ■ (£, • ) of rank r. Then the chromatic number of Jl" is
A* &
A set F c £is independent if. and only if. it contains no circuits. Therefore,
a partition Mi, Aa Av) is a colouring of hypcrgraph //M if, and only if.
sets Alt A7 A, arc independent. Thus, *(//") - /»(.!/), and the corollary
follows from Corollary I.
Q.E.D.
The above results easily yield several theorems for graphs that were proved
earlier by direct but very involved methods.

MATROIDJ 491
Application I (Theorem of Tutte A961]). The edge set of a simple con-
connected graph G — (X, E) contains k painvise disjoint spanning trees if, and only
if, for each partition of X, the number m^ ) of edges that join two distinct
classes of the partition satixjies
»»„(. )>*(|*| - I).
I. If there exist k spanning trees //t, H3 Hk thai arc pairwise edge-
disjoint in G, then Tor each partition of the vertices,
wM(( )>| | - I </- 1,2 k).
Therefore,
)> 2>H.< )>fc(i l-D.
ii
which is the required condition.
2. Suppose that the condition of the theorem is satisfied. Consider the
miitroid M — (£.. ) where is the family of forests F,. F2 F, of graph
G. Let r(A) denote the rank of .V. If A c /; defines a partial graph of G with
p connected components that constitute a partition - (,V|. Xt Xv)
of X, then
r{E) - r(A) - (w - I) - (// - p) - p - I - |. | - I .
Therefore, from the condition of the theorem, it follows that
\E-A\> mi6(. ) > A(|. | - I) - *(/■(/•) - /■(/«)).
Hence.
r.
Thus, from Corollary 2 to Theorem 7, there exist k disjoint spanning trees
in G.
Q.E.D.
Application 2 (Theorem of Nash-Williams [1964]). The edges of a simple
graph G ■ {X, E) can he coloured with k colours such that no cycle has all its
edges with the same colour if ami only if, for each set A <= X, the number
'»<,M. A) of edges with both endpoints in A satisfies
mG(A,A) < k{\A\ - I)
In other words, the chromatic number of the hypergraph G° formed con-
consisting of the cycles of G is
ll>l

492 HYPERORAPIU
1. Suppose that the edges are coloured with k colours 1,2, ...,k. Let
mt(A, A) denote the number of edges with colour / that have both endpoints
in A. These edges form a forest. Therefore, m,{A, A) < | A \ - 1. Hence,
mo(A,A) - m,M,y<) + - + mk(A.A) *k{\A\-\).
and the result follows,
2. Conversely, suppose that the condition of the theorem is satisfied. Con-
Consider the matroid (£,. ) formed by the family of forests of graph G. Let r de-
denote the rank of this matroid. If the partial graph (,V, F) of G generated by
F c £ has exactly p connected components (Xlt F,), {Xa, Fa) (Xp, Fp)
that are not isolated vertices, then
- | F, | > *(| X, | - 1) - ,»<;(*„ X,) > 0.
Hence,
Ar(/-) - I/■'| - £ {kr{Fl)-\Fl\)*0,
/•i
and
max M-tt4 •
Thus, from Corollury 4, it is possible to colour the edges of G in k colours
such that no cycle has only one colour.
Q.E.D.
Appmcation 3. If C is a simple graph with maximum degree h. then it is
possible to colour its edges with [ ' ] +1 colours such that no cycle has all its
edges with the same colour.
Let C - (X. E) be such a graph, and let A c X. If | A | > 1. For a eA,
let A - A {a}. Thus
1 mc(A. A) - -1- (m<,G,1) + n^Q. a)) <
A\ - I
\\
ur 2
From the Nash-Williams theorem (Application 2), it follows that

MATROIDJ 493
Therefore, it is possible to colour the edges of G with [ 1 +1 colours such
that no cycle of C has all its edges with the same colour.
Q.E.D.
4. Minimum weight basis
A maximal independent set of a matroid M is called a basis. Associate with
each clement e of matroid M a positive number p(e) called the weight ofe. The
weight of a basis F of M is defined to be
We shall consider the problem of determining a minimum weight basis of a
matroid.
Example I. Minimum tree in a weighted graph.
Let C - (X, E) be a connected simple graph. To each edge e associate a
"length" p(e) > 0. The minimum tree problem is to find a spanning tree
// - (X, F) of G whose totul length
Pin- 2 Pie)
is as small as possible.
This problem due to G. Choquct [1938] and J. B. Kruskal [I9S6] has nu-
numerous applications. For example, n cities in the Benelux countries must be
connected by canals, and the total length of the canals must be minimized
(and no branching is allowed except at the cities). Since the system of canals
must be connected and since its length must be minimum, no cycles are
allowed. Thus this problem becomes a minimum tree problem.
The minimum tree problem reduces to finding a minimum weight basis for
the matroid (£, ) formed by the forests of G. (See Example 3, § I.)
Example 2. The following problem studied by Chein [1970] also reduces to
a minimum tree problem: Consider a connected simple graph G - (X, E)
with a positive length />(<') associated with each edge e. and a set S c X. Find
a tree in C (not necessarily spanning) that contains all the vertices of S and
that has a minimum total length.
When all lengths are equal to I. this problem can be solved by treating
separately each of the connected components Xlt X3,.... Xp of the subgraph
Gs, that respectively have spanning trees (Xl% Ft), (Xa, F2) (Xp, Fp). and
by finding a spanning tree Fo of minimum weight in the graph obtained from
G be contracting *,. X3 Xp. Then. Fo u F, u ■•• \j Fp is the required tree.

494 HYPERORAPIIS
Let h - (p|, e2 e,) arid F' - (e[, pJ, .... e'r) be two ordered bases of
malroid M - (£,. ); the elements of these r-tuplcs arc indexed such that:
P(e\)
Let F < F' if, and only if. there exists an index k < /• such that
p(fi) - Pd'|) for i < k
This order relation -< on the set of bases is called a lexicographic ordering: if
the "weight" of a letter is its rank in the alphabet, then the relation F ■< F'
between two words Fand F' indicates that word F appears before word F'
in the dictionary.
Proposition I. If F — (<•,, fa, .... er) is an ordered bash of a malroid
M tm (E,. ), then the following three conditions are equivalent'.
A) F is a minimal basis with respect to the lexicographic ordering, a.
B) each ordered basis F' - (e[. c'a e'k). satisfies p{e,) < />«) for all /.
C) F is a minimum weight basis.
(I) B). Otherwise, let F be a minimal basis, and let /■" be a basis that
docs not satisfy B), Let A- be the first index such that p(ek) > pie\,)
has rank r(A) > k, and consequently there exists an index / < A- such that the
set
is an independent set of cardinality k. Moreover, since / < A\ we have:
pd'\) < p(ek) < p(ek).
Let ii be the basis that contains Fo with | B\ - /'. Thus, li < F. which
contradicts the minimality of F.
B) C). This is obvious.
C) A). Let F be a minimal basis with respect to the lexicographic
ordering, and let F' be a minimum weight basis. Since (I) implies B). basis F
satisfies B), and
/>(<•() < /'(<•!) (' ■ '«2 r).
Since F' has minimum weight, it follows that
/'(«,) -/H<'i) ('- 1.2 r).

MAT HOI US 49S
Thus /' -< /•"; since /"is minimal with respect to the ordering a. it follows
that /-" is minimal with respect to the lexicographic ordering a.
Q.H.D.
Algorithm I. To construct a minimum weight basis, first choose An clement
c, of minimum weight, and let F, m {i\ }. Then choose any clement ca with
ca F, and /•', u { e2 }e- . that has minimum weight: let Ft - Ft u {c3 }
etc.
After /• steps, a basis F, is determined. Since Fr is minimal with respect to
the lexicographic ordering, it is a minimum weight basis by Proposition I,
Dual algorithm. If Fis a basis of matroid .1/ • (£,. ). then its complement
E - /' is called a cohasis. Clearly, the cobascs of .1/ are the bases of some
matroid M* called the dual matroid of M. If .1/ is a matroid whose bases
arc the spanning trees of a connected graph, then the cobascs of M arc the
cotrccs of this graph.
Instead of constructinga minimum weight basis for matroid M with weights
/>(<')< we could construct the maximum weight cohasis or construct n minimum
weight basis for the dual matroid A/* with weights /»•(<')- /><, - /»(().
where p0 is it sufficiently large positive number such that p*W) > 0 U e E).
To construct a maximum weight cobasis for V, first remove from E an
clement c, of maximum weight. I ct Ex — E — {et}. Then, remove an cle-
clement e.u e E, that has maximum weight. Set £'a - £| - { ra} also has rank r.
After »/ - /■ steps, a maximum weight cobasis £ - /;„_, and a minimum
weight basis £„. have been found.
For the special case of constructing a minimum length spanning tree of a
graph G. the above algorithms can be improved. First, we need the following
proposition;
Proposition 2 (Roscnstichl [1967]). Let (i - (A'. U) be a connected graph
with a length />(//) associated wuth each arc u. Suppose that the lengths are all
different. Then, the tree of minimum length is unique, A necessary and sufficient
condition that are r belongs to this trce(X, V) is that there exists an elementary
eacycle to in C such that
A) p(v) m min p(n).
Another necessary and sufficient condition that arc r belongs to tree (X, V)
is that far taih cycle //,
B) p(v) < max p(u).
iff M

496 HYPfRORAPHS
1, The minimum tree is unique. Otherwise, if V~(vx.v vr) and
W - (h-,. vra ur) were both minimum trees, then from Proposition 1.
/•(p.) < pi»'i) < P(vt) (' - J • 2..... r).
This contradicts the assumption that the arc weights are all different.
2. If arc r belongs to the minimum tree (X. V), then v separates two sets A
and X - A. clearly.
p(v) - min
(Otherwise, there would exist an arc r' r in the minimum tree chosen by the
algorithm I to link sets A and X — A.) Therefore, there exists a cocycle w(A)
that satisfies (I).
3. If arc <- belongs to a cocycle w that satisfies (I). then r also belongs to an
elementary cocycle w(A) that satisfies (I). and r would be the (irst edge ofw(A)
to be chosen by Algorithm 1 to construct the minimum spanning tree V.
Therefore, r e V.
The proof of condition B) is left to the reader.
Algorithm 2 (Kruskal [1956]. Sollin [1962]). A sequence Vu Va,... of
forests can be successively formed in the following way:
(I) Let Vx — 0; forest (X, V^) consist of// isolated vertices,
(II) If forest (-V, V,) has an isolated vertex *, then join a- to its closest
neighbour; i.e., if u,, i is the arc of minimum length incident to x, then let
^/M - ^1 U («,♦■}•
(III) If forest (X. V,) has no isolated vertices and is not connected, then
join a connected component to its closest connected component.
(IV) If (,V, V,) is connected, then it is a spanning tree and, clearly, this tree
has minimum length hy Proposition 2.
Algorithm 3 (P. Rosenstichl A967]). Starting with a cocycle basis {co1, w3,
....w"} remove from each of these cocycles its minimum weight arc. Then,
proceed as in Algorithm 2.
Dual algorithm. Starting with a cycle basis {/('. ft2 yt1} (for example, if
G is planar, the contours of the bounded faces), remove successively from
euch cycle an arc with non-maximum weight. Continue until no cycles remain.
EXERCISES
1. Show (hat (he set y of circuits of a mutroid satisfy the following:
A) 5' c 5, say implies 5' - S
B) S,S"cy.S S.aeSnS1 Implies that there exists an So with Se <= S^
S'-ia).

MATROtDS 497
Conversely, show that if a family of sets wit Mies (I) and B). then the family is the set of
circuits of some matroid.
2. Let Kn denote the complete graph with « vertices, nnd let Kt denote the hypcrgraph
consisting of the cycles of Kn. Show that the Nash-Williams theorem implies that
Show how to construct a colouring of the edges or graph A', in "I colours such that
no cycle has all its vertices with the same colour.
3. Let Kn, „ denote the complete bipartite griph. Show that
4. Let AS be a matroid on set E. Show that if • Is the family of Its bases then
IE - Bl Be I is the set of bases of matroid A/.
5. Let G ■ (A". O be a t-graph. and let flu be a subset of A' with cardinality k. Let j/ ■
{ F(x) \j {x)Ixs X - Ho\. For a set H c A' with cardinality k, show that the following
conditions arc equivalent:
A) Sets Bund Bo can be joined together by k paths that arc pairwisc vertex disjoint.
B) The set X - B is a set of distinct representatives of family • (sec Section 2).
6. Use Fxcrciscs 4 and i to show the following result;
Let G ■ (A". T) be a t-graph. and let Bo <= X. Then the sets lie Xsuch that | B | ■
| Ho | and such that there exist j B,, | pairwisc vertex-disjoint paths Joining B and B<, con-
constitute the buses of a matroid on X.
(II. Perfect, Applications of Monger's Theorem.
Math. Anal. AppL 22. 1968. pp. 96-111.)
7. Let G ■ (A', /') be a transitive I-graph, and let ■ be the family of subsets F of A'such
that P'HX - /■') = /•". Show thut is a matroid on X.
8. Let G be a multigraph. Let be the family of subsets of the edge set that do not con-
contain an even elementary cycle and that do not contain two edge-disjoint odd elementary
cycles. Show that Is a matroid on E. (A. Duchamp. Thesis, Caen A971])
9. In the (directed) graph G - (A'. U). let be the family of subsets Uo of U ■
{iii, ut «„ } that do not contain an "anti-circuit" (a simple even cycle whose arcs
arc alternatively directed).
Show that is a matroid on U. Determine its rank. (A. Duchamp)
Hint: Consider the graph // ■ ( Y. V) with Y - ( a. h >. and with m arcs vi,e» vm
directed from a to b. Consider the "product" G 9) H. i.e. the graph on X x Y with an
arc from (x. y) to (*'. /) denoted by wk. if (x, x') ■ uk and {y, y) ■ v».
Clearly, each anti-circuit of G is a cycle of G ® //. and each cycle of G ® N is an anti-
circuit of G.
10. Show that if (£. ) and (E, ') are two matrolds on £ whose ranks arc respectively
r and r\ then
max | S | - min [r(A) + r'(E - A)].
*c n, ' ilcl
(This is the "Matroid Intersection Theorem" of J. Edmonds).

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CHAPTER 8: r-MaU-liings
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S. Rudianu, Notes sur I'cxistcnce ct I'unicilc du noyau d'un gruphc. Revue
Francalse Rech. Opirat., 33. 1964. 20 26 and 41. 1966. 301 310.
C. A. B. Smiih. "Gruphs and Composite Gumcs". A Seminar on Graph Theory
(F. Hurary-L. Bcinckc). Holt Rinchart. Winston. New York, 1967.86 III.
R. Si'RAaut, Bcmcrkungcn ubcr cine spczicllc Abclschc Gruppc. Math. Z., 51,
1947, 82-93.
L. P. Varvak, Existence or a Kernel in a Product (Russian), Ukr. Mat. Zh., 17.
No. 3, 1965, 112-114.
L. P. Varvak. Gumcs on a Sum of Graphs (Russian). Kiber., 4. No. I, 1968,
63 66.
V. G. Vizinc), A Bound on the Extcrnul Stability Number of u Graph. Doklady
A. N. 164. 1965,729-731.
V. G. Vizinq, The Cartesian Product of Graphs. Vych. Sis., 9. 1963, 30 43.
P. M. Wec-hsci.. The Kroncckcr Product of Gruphs. Proc. Am. Math. Soc, 13,
1962, 47 52.
C. P. Wr.i.rr.R, The Theory or a Class of Games on a Sequence or Squares, in
Terms of the Advancing Operation in a Spcciul Group. Proc. Kon. Netlerl.
Akad. N. (sdric A), 57. 1954, 194 198.
N. Y. WiTHorr, A Modification of the Game of Nim, Nleuw Arch, roor WUkiuule,
7, 1907, 199-202.
CHAPTER IS: < hromutlc Number
C. BfcKCE, Lcs problcmcs dc colorations en Thcoric des Gruphcs. Publ. hist. Stat.
Vniv. Paris, 9, I960, 123-160.
G. Berman, W. T. Tutte, The Golden Root of u Chromatic Polynomial. J.
Combinatorial Tlieory. 6, 1969, 301 302.
G. D. HiRKiiorr, A Determinant Formula for Coloring a Map. Ann. Math., 14,
1912,42 46.
G. D. RiKKitorr, D. C. Lewis, Chromatic Polynomials, Trans. Amer. Math. Soc,
60. 1946. 355 451.
R. L. Brooks, On Colouring the Nodes of a Network. Proc. Cambridge Phil. Soc,
37. 1941, 194 197.
V. Chvatal, A Note on Coefficients of Chromatic Polynomials, J. Comblnat.
Theory, 9, 1970, 95 96.
V. Chvatal, J. Komi 6s, Some Combinatorial Theorems on Monotonicity.
Canad. Math. Dull., 14 B), 1971.

KCFI'KI'NCES 313
G. A. Dikac, A Properly of 4-chromatic Graphs and Some Remarks on Critical
Graphs. J. London Math. Soc, 27, 1952, 85 92.
G. A. Dikac, Mup-colour Theorems. Can.J. Math., 4, 1952, 4K0 490.
G. A. Dikac, "Valcnce-variccy and Chromutic Number". IVlss. 7.. Martin-
Luther Univ. Halle-Wittenberg, 13, 1964, 59 63.
P. Eri>os. G. Szikirm. A Combinntorial Problem in Geometry, Compositio
Math., I. 1935,463-470.
A. Ehrera. Exposd hiscoriquc du problemc des quuirc coulcurs. Periodlca di
Mat., 7, 1927.20 41.
A. Ekkika, Unc contribution an problcmc des quuirc coulcurs. Hull. Soc. Math.
France, Si, 1925,42 55.
H. J. Finck. Obcr die chromatischen Zahlcn cincs Graphcn und seines Komplc-
ments. I, II, 1966. Milteiluiig aus dem histitut fiir Mallieinalik, llmenau.
J. C. Fourniek, Eflct dc la contraction d'unc urttc sur Ic genre d'un graphc. C. R.
Acad. Sc. Paris, 270. 1970, 743 744.
P. Franklin, Note on the Tour Color Problem. J. Math. Phyx, 16. 1938. 172 184.
T. Gai.i.ai, Krilischc Graphcn, I and II. Publ. Math. hist. Hungarian Acad. ScL,
A 8, 1963, 165 192; 9. l%4, 373- 395.
T. Gallai, "On Directed Paths and Circuits". Tlieory of Graphs, Tihany (P.
Lrdos, G. Kutonu, cd.). Acndcmic Press, New York. 1968. 115 118.
M. (jk6t/sch, Fin DrcifarbcnsHl/ fur dreikreisfreic Net/.e uuf dcr Kugcl. Wls\. Z.
Martin Luther Unit'., Halle-Hittenberg. Math. Natiirwlss. Reihe, 8. 1958.
109 119.
B. GkUnraum, Grolzschc's Theorem on 3-colorings. Michigan Math.J., 10, 1963,
303 310. -
R. P. Gupta. Studies in the Theory of Graphs. Thesis, Tata Inst. of Fund. Res.
(mimco.), Bombay. 1967.
R. P. Gupta, Bounds on the Chromatic and Achromatic Numbers of Complemen-
Complementary Graphs. Miinco. scries No. 577, Univ. of North Carolina hi Chapel
Mill, 63, April 1968.
II. IIadwiger, Obcr cine Klassifikntion dcr Streckcnkomplcxc. Vierte Natiir-
forsch. Ces. Zurich, 88. 1943, 133 142.
G- Maj6s, Obcr cine Konstruktion nichl w-fiirbbarcr Graphcn. Mm. Zeitschr.
Martin I nlher Unit'. Halle-Wittenberg, A 10, 1961, 116 117.
D. W. Mai l. J. W. Siry and B. R. Vandekslke, The Chromatic Polynomial of
the Truncated Icosahcdron. Proc. Amer. Math. Soc, 16. 1965. 620 628.
S. T. Hedetkifmi, Honiomorphism or Graphs and Automata. Tech. Rep., The
Univ. of Michigan, Ann Arbor, 1966.
C. Hi uciienne, Sur Ic critcrc dc chromaticitc dc Miij6s-Orc. Dull. Soc. Roy. Ltige,
12, 1968, 10 13.
M. R. Iyer, V. V. Menon. On Coloring the n x n Chessboard. Am. Math.
Monthly, 73. 1966.721 725.
E. L. Johnson, A Proof of the Four-Coloring or the Edges or a Regular Three-
Degree Graph. O. R. C. 63 28 (R. R.) Mini, report 1963. Operations Re-
Research Center, Univ. of Calif.
G. Kki.wi.ras, Pcut-on former un rcscau nvee des purtics finics d'un ensemble
dcnombrnblc. C. R. Acad. Sc. Paris, 222, 1946, 1025 1026.
L. S. Mf.lnikov, V. G. Vi/ino. New Proofs ol" Drooks Theorem. J. Combinatorial
Vieory, 7. 1969. 289 290.

314 urn minces
E. A. Nokdhaus, J. \V. Gaddum, On Complcmcntiiry Graphs. Ann. Math.
Monthly, 63. 1956. 175 177.
O. Ore. The Four Color Problem. Academic Press. New York. 1968.
F. I*. Kasufy, On a Problem of Formal Logic. Proc. loiuion Math. Soc. 30. 1930,
264 2Kfi.
I). K. RAY-CiiAumiUKt and R. Wilson. Solution of Kirknian's Schoolgirl
Problem, Proc. Symposium Pure Moth., 19, A.M.S., Providence, 1971,
187 203.
R, C. UtAi), An Introduction to Chromatic Polynomials../. Combinatorial Theory,
4. 1968,52 71.
G. KiNUEL, FUrbuiiKsproblt'ine tmf Fh'lchen nml Graphen. Her I in, 1959.
G. KiNCiLL, J. W. T. Ynrsos, Solution of the I lea wood M.\p-coloring Problem,
./. Comb. Theory, 7. 1969. 342 363.
G. C. Rota, On the Foundations of Combinatorial Theory, I, W. I. G.. 2. 1964,
340 368.
H. Roy, Nomhre chromatique et plus longs chemins d'un grnphc. Kerne AFIRO,
1, 1967. 127 132.
M. Schauih.e, " Bcmcrkungcn zu cincm Kanicnfiirbungsproblcm". lieltr&ge :nr
Graphentheorie, Leipzig, 1968. 137 142.
W. T. Turrt, On Chromatic Polynomials and the Golden Ration. J. Combina-
Combinatorial Theory (to appear).
K. Wacjnek. Bcmcrkungcn zu lladwigcr's Vcrmutung. Math. Ami., 141. I960,
433 451.
K. WacnI'K, Beweis einer Abschwachunjj tier H.idwigcr Vcrmutung. Math. Ann.,
153, 1964, 139 141.
H. WmrNFY, A Logical Fxpansion in M.ithem.itics. Hull. Amer. Math. Soc, 3K,
1932, 572 579.
II. Whitney, The Coloring of Graphs. Ann. o Math., 33. 1932. 688 718.
C. t. Winn, A Case of Coloration in the Four Color Problem. Am. J. Math., 49,
1937,515-528.
A. A. Zykov, On Some Properties of Linear Complexes (Anur. Math. Soc.
Transl. No. 79, 1952). Mat. Sb., 66. 1949, 163 188.
CHAPTFR 16: Perfect Graphs
O. Annuli. On the Sum of Graphs. Revue Fr. Rech. Opiratlonnelle, 33. 1964,
353 358.
M. II. MrANORLW, On the Product of Directed Graphs. Proc, Am. Math. Soc,
14. 1963.600 606.
C. Bfrof, Lcs problenics dc coloration en thcoric des graphes. Pnbl. Inst. Stat.
Univ. Paris, 9. I960, 123 160.
C Bfkcc. " Farbung von Gniphcn, dcrcn sa'mtlichc bzw. dcrcn ungcradc Kreisc
starr sind". Wissenuhajtllchc Zeitung, Martin Luther Unii\, Halle Witten-
Wittenberg, 1961, 114.
C. BrRCiE, Sur line conjecture relative au probldme ties codes optimanx, coinm. 13
erne asscmblec gcncralc dc I'URSI, Tokyo, 1962.
C. Hicrc.e, "Perfect graphs". I. Six Papers on Graph Theory, Indian Statistical
Institute, Calcutta, 1963.

KEfliKENCCS SIS
C. Bi roe. " Unc application dc la thcoric dcs Gruphcs a un prohleme dc codngc".
Automata Theory (K. R. Caiancllo cd.) Ac;ulcmic I'rcss, I omlon, 1966,
25 34.
C. BiiRCit, "Some Classes of Perfect Graphs". Chap. 5. Graph Tlteory and
llieoretical Physics (F. Marary, cd.). Acuclcmic I'rcss, New York, 1967.
C. BuuiK, "The* Rank of a Family of Sets und Some Applications to Graph
Theory". Recent Progress In Combinatorics (Fulic, cd.). Academic Press,
New York, 1969,49 57.
C. HiRCiL, M. Las V iron as, "Sur un thcoremc du type K6nig pour hyper-
graphes". Annals New York Acad. Sc, 175, 1970, 32-40.
I*. Camion, Matrices totalement unimodiilnires et proble'mescombinatolres.Thcsk,
University Libre dc Bruxcllcs, 1963.
Y. Chao. On a Problem of Claude Bergc. Proc. Am. Math. Sac., 14, 1963, 80 82.
D. R. Fui.KrKSDN, Notes on Combinulori.il Mathematics; Anli-Ulocking Poly-
hedra (mimeo). RM 620/l-PR, February 1970.
T. Gai i ai, Grnphcn mil triangulicrbaren ungcraden Viclcckcn. Magyar. Tud.
Akatl. Mat Kumtd Int. Kiizl., 7, 1962. 3-36.
T. Gallai, Transitiv oricnticrbarc Graphcn. Ada Math. Acad. Sc. Hangar., 18,
1967, 25 66.
A. Ghouii a-Houri, Sur la conjoncture de Berge. Mini. Scminairc Probl. Com-
binnt. IMP, I960.
A. Ghouila-IIouui, Carnctc'risalion dcs Graphes non oricntc's dont on pcut
oricntcr les nrilcs dc mnnicrc ti obtcnir Ic graphc d'unc relation d'ordre.
C. R. Avail. Sc. Path, 254. 1962. 1370 1371.
A. Ghouila-IIouri, Carnct<irisation dcs matrices touilcmcnt unimodulaircs. C. R.
Acad. Set. Parts. 254. 1962. 1192 1193.
P. C. Gii.mori, A. J. I Iofjman, A Characterization of Compambility Graphs nnd
of Interval Graphs. Camid.J. of Math., 16. 1964. 539 548.
A. Hajnai, J. SurAnyi. Obcr die Auflosung von Grnphcn in vollstandigc Tcil-
graphcn. Ann. Univ. St. Budapestinenuls, 1. 1958, 113-121.
G. Maj6s. Obcr cine Art von Graphcn. Im. Math. Nachrichten. II, 1957.
A. KoT7ic, Pair Hajos graphs (Slovakinn). Casopispesi. mar., 88, 1963, 236-241.
C. G. LFKKEKKiiKKKK. J. Ch. Boi.and, Representation of n Finite Graph by a Set
of Intervals on the Real Line. Fund. Math., 51. 1962, 45 64.
Ljunini, Note on n Problem of Claude Dcrgc (Russian). Sibir. Math. 7.h. {.Journal
Math, de Siberle), 5. 1964. 961 963.
L. LovAsz, Normnl Hypcrgrnphs nnd the Perfect Graph Conjecture. Discrete
Malh.,2, 1972,253-267.
D. J. Miilfk. Categorical Product of Graphs. Camid.J. Math., 20, 1968, 1511-
1521.
C. F. Picard, "Lntticoid Product, Sum and Product of Graphs". Combinut trial
structures and their applications, Gordon and Breach, New York, 1970, 321-
322.
C. F. Picakd, Distributivitii d'op^rntions sur les grnphes. C. R. Acad. Sc. Paris,
270, 1970. A. 1219 1221.
A. Pultr, "Tensor Products on the Category of Graphs". Combinatorial struc-
structures and their applications, Gordon and Breach, New York, 1970.327-329.
H. Sachs, "On the Bcrgc Conjecture Concerning Perfect Graph". Combinatorial

SI 6
structures and their applications, Ciordon and Breach, New York, 1970,
377-384.
C. E. Shannon. The Zero-error Capacity of u Noisy Channel. Comp. Information
Theory, IRE Trans., 3, 1956, 3 15.
L. SurAnyi, The Covering of Graphs by Cliques. Studio Sc. Math. Hung.. 3.1968,
345 349.
V. G, Vizincj, The Cartesian Product of Graphs. Vy(. Sis., 9. 1963. 30 43.
l\ M. WtcHSKL, The Kroncckcr Product of Graphs. Proc. Am. Math. Soc, 13,
1962.47-52.
E. S. \Voi.k, A Note on the Comparability Graph of a Tree. Proc. Am. Math. Soc,
16. 1965, 17 20.
CHAPTFR 17: llypcntraphs and their Duals
A. Andrlatia, Sur les graphes finis qui sont des udjoints (Italian). 1st. lomhardo
Ac. Sc. Lett. Rend., A 98. 1964, 133 156.
L. W. Rr.iNi-.KE, "On Derived Graphs and Digraphs". Beitriige zur Craplientheorie
(II. Sachs. II. J. Voss, II. Walt her. cd.). Tcubncr, 1968, 17 23.
C. Bf.rge, Le graphe adjoint d'nn graphe. Mimco. Scininairc sur les problcmcs
combinatoircs. I. II. P. 1959.
C. BrROC, "The Rank of a Family of Sets and Some Applications to Graph
Theory". Recent progress In Combinatorics (W. Tuttc, cd.). Academic
Press, New York. 1969, 49-57.
C. Beroe, R. Rado, On Isomorphic Hypcrgraphs, and on Some Extensions of
Whitney's Theorem to Families of Sets. J. Comb, liteory R. 13. 1972, 226 241.
J. BosAk. "The Graphs of Semi-groups" Proc. Symposium Smolenice, Prague
1964, 119 125.
R. G. Busackfr. T. L. Saaiy, Finite Graphs and N'tworks. McGraw-Hill, New
York. 1965.
G. Chartrand, Graphs and Their Associated line-graphs. Thesis, Michigan State
Univ., 1964.
K. Culik. "Applications of Graph Theory lo Mathematical Logic and Lingui-
Linguistics". Proc. Symp. Graph Theory, Smolenice, 1963, 13-20.
P. Erdos, On Extremal Problems of Graphs and Generalized Graphs. Israel J.
Math.. 2. 1964. 183 190.
P. Erdos, A. W. Goodman, L. Posa. The Representation of a Graph by Set
Intersections. Canal, J. Math., 18. 1966, 106 112.
P. C. Gilmore, A. J. Hoffman, A Characterization of Comparability Graphs.
Canad. J. Math.. 16. 1964, 539-548.
B. GrOnbaum, "Incidence Patterns of Graphs and Complexes". The Many
Facets of Graph Theory (G. Chartrand, S. F. Kapoor, cd.), Springer
Verlag, 1969, 115 128.
B. GrOnbaum, Convex Polytopes, Wiley, New York, 1967.
A. Gyarfas. J. Lrhfl, "A llclly-typc Theorem on Trees", Combinatorial Theory
and its Applications (Erdos, Rcnyi, V. T. S6s, cd.), North-Holland Publ.
Co., Amsterdam-London, 1970.
R. K. Guy, E. C. Milner. Graphs Defined by Coverings of a Set. Ada. Math. Ac,
Sc. Hung,. 19. 1968. 721-735.

RE EMINC13 SI7
F. Hakary, Graph Tlieory. Addison-Wcslcy. Reading, 1969.
P. Hill, J. NeSetril. Graphs nnd A-socictics. Canad. Math. Bull., 13. 1970, 375-
381.
C. HeucHENNE. Sur ccrtaincs corrcspondanccs enlrc graphes. Bull. Sac. Roy. Sc.
Liege, 33. 1964.743-753.
A. J. Hoffman, On the Uniqueness or the Triangular Scheme. Ann. Math. Stat.,
31, I960, 492-497.
A. J. Hoffman, On the Line Graph or the Complete Bipartite Graph. Ann. Math.
Star. 35, 1964, 883 885.
J. Krausz, Demonstration nouvcllc d'un thlorcmc dc Whitney sur les rcscuux.
Mat. Fiz. Lapok, 50. 1943, 75-85.
G. Kreweras, "Counting Problems in Dcndroids". Combinatorial structures and
their Applications, Gordon nnd Breach, New York, 1970, 223-226.
W. Kuish nnd N. Saucr, "On I ho Existence or Certain Minimal Rcgulur w-
systcins with Given Girth". Proof Techniques In Graph Tlieory (F. Harnry,
cd.), Academic Press, 1969, 93-101.
L. LovAsz, "Graphs and Set Systems". Beitrtige zur Graphentheorle (H. Sachs,
H. J. Voss, H. Walther. cd.), Tcubncr, 1968, 99-106.
V. V. Menon, The Isomorphism between Graphs and their Adjoint Graphs.
Canad. Math, Bull,. 8. 1965, 7-15.
V. V. Menon, On Repealed Interchange Graphs. Am. Math. Monthly, 73. 1966,
986-989.
J. Moon, On the Line-graph of the Complete Bigruph. Ann, Math, Stat., 34. 1963,
664-667.
A. Ramachandra Rao, Some Extremal Problems mil Characterizations In the
Theory of Graphs. Thesis. I. S. I., Culcuttn, 1969.
D. K. Ray-Chaudhuri. Characterizations of Unc-grnphs. J. Comb. Theory, 3,
1967, 201-214.
F. S. Roberts, " Indifference Graphs". Proof Techniques in Graph Theory,
Academic Press, 1969.
F. S. Roberts, On the Boxity and Cubicity or a Graph. Recent Progress In
Combinatorics, (Tutte, ed.) Ac. Press, 1969, 301 310.
F. S. Roberts, J. H. Spencer, A Characterization or Clique Graphs. J, Comb.
Theory, 10. 1971, 102 108.
A. C. M. van Roou. II. S. Wii.f, The Interchange Graph of a Finite Graph. Ada
Math. Sc. Hungar,, 16, 1965, 262-269.
G. Sabidussi. Graph Derivatives. Math., 2, 76, 1961, 385 401.
S. Seshu, M. Reed, Linear Graph and Electrical Network, Addison Wesley.
1961.
S. S. Shrikhande, On n Characterization of the Triangular Association Scheme.
Ann. Math. Stat,, 30. 1959, 39 47.
H. Whitney, Congruent Graphs and the Connectivity of Graphs. Amtr. J. Math.,
54, 1932, 150-168.
CHAPTFR 18: Transversals
L. W. Blineke, "A Survey of Packings nnd Coverings of Graphs". The Many
Facets of Graph Theory (G. Chnrtrund, S. F. Kupoor, ed.), Springer
Vcrlag, 1969,45-53.

518
C. Bi roe, "The Rank of a Family of Sets and Some Applications to Gr.iph
Theory". Recent Progress in Combinatorics (W. Tulle, cd.), Academic
Press, 1969. 49 57.
J. Edmonds, A Minimum Flcmcnt-covcring from u Finite Class of Weighted Scls.
April meeting of A MS. April 1961. 578 592.
P. ERDtfs, T. Gallai, On Ihc Minimal Number of Vertices Representing the
Fdgcs of j Graph. Magyar Tmi. Akad, Mat, Kutato. 6. 1961. 89 96.
D. R. Fulkfrson. H. J. Rvsfr. Widths and I Icights of @. I) Matrices. Cwail, J.
of Math,, 13. 1961.239 255.
E. GRiNMiRa, Ya. Damint, Some Properties of Graphs Containing Circuits.
Late. Mat. Ezh.. 1965.65 70 (Russian).
R. K. Guy, A Coarseness Conjecture of Erdos. J, Combtnat. Theory, 3, 1967.38
42.
R. K. Guy, L. W. Buneke, The Coarseness of the Complete Graph. Canad. J.
Math,, 20, 1968, 888 894.
F. Jaecfr. C. Payan, Determination du nombrc maximum d'arclcs d'un hypcr-
graphc T-criliquc. C.R. Acad, Sc. Park 273. 1971, 221 223.
M. Las Vfrcinas, Transvcrsalcs disjointcs d'une famillc d'cnscinblcs (to appear in
Discrete Maihematici).
E. L. Lawlfr, Covering Problems: Duality Relations and a New Method of
Solution. SIAMJ. Appl. Math,. 14. 1966, No. 5.
K. Magiiout, Sur la determination des nombrcs de slabilite el du nombrc chro-
matiquc d'un grnphc. C. R, Acad. Sc. Paris. 248, 1959, 3522-3523.
D. K. Ray-Ciiaudhuri, An Algorithm for a Minimum Cover of an Abstract
Complex. Canail.J. Math., 15. 1963, II 24.
B. Roy, Algebre moderne et Thiorie des graphes, II, Dunod, Paris, 1970. Chap.
VI, B.
N. Saufr, On the Factorization of //-graphs. Res. paper. The Univ. of Calgary,
No. 67, 1969.
V. E. Tarakanov, The Maximal Depth of a Class of @. l)-malriccs. Mat. Sb.,
75, 1968, 4 14.
CHAPTFR 19: Chromatic Number of a HypcrRraph
C. Bcroe, Sur certains hypergraphes gcncralisanl les graphes bipartitcs, Cow;W/w-
torial Theory and its applications, Balatonfured (P. Erdos, A. Kcnyi, V. T.
Sosed.), North-Holland Publ. Co., Amsterdam-London, 1970, 119 133.
G. Chartrand, D. P. On i.i-r. S. I 1i:detniimi. A Generalization of the Chromatic
Number. Proc. Cambridge Phii. Soc. 64. 1968, 265 271.
F. R. K. Chuno. On the Ramsey number N C. 3 3; 2). Discrete Math.. 5. 1973.
317-322.
V. Chvatal, Hypergraphs and Rumscyian Theorems. Proc. Am. Math. Soc., 27,
1971,434-440.
V. Chvatal, The Smallest Triangle-free 4-chromatic 4-rcgular graph., J. Com-
blnat. Theory, 9, 1970, 93-94.
B. Descartes, A Three-colour Problem. Eureka, April 1947 (solution, March
1948).
P. ErdOs, Some Rcinurks on the Theory of Graphs, Bull. A.M.S.. 53, 1947, 292-
294.

SI9
P. ErdOs. R. Rado, A Construction of Graphs without Triangles Bearing Pre-
ussigned Order and Chromatic number. J. London Math. Soc, 35, I960,
445-448.
P. Erd5s, Ciiao Ko, R. Rado, Intersection Theorems Tor Systems or Finite Sets.
Quart. J. Oxford, 1961, 12, 313-320.
P. Erdos, A. IIajnal, On Chromatic Number of Graphs and Set-systems. Ada
Moth. Aaut. Sc. Hungaricae, 17, 1966. 61 99.
P. Erd6s, A. S/eklres, A Combinatorial Problem in Geometry. Compositio
Mith.,1, 1935,463 470.
C. Frasnav, Sur des functions d'entiers sc rupportant au thcorcmc de Ramsey,
C. R. Acad. Sc. Paris, 256. 1963, 2507-2510.
L. Geri-ncsfr, Srinczcsi Problemakr6l. Mat. Lapok, 16, 1965, 274-277.
R. G. Giraud, Unc majorution des nombres de Ramsey binuires-bicolores, C.R.
Acad. Sc. Parts, 266. 1968, A 394 396.
R. Graham, On Edgewise 2-colored Graphs with Monochromatic Triangles and
containing no Complete Hexagon../. Comb. Theory, 4. 1968, 300-301.
R. E. Grfenwood, A. M. Gi lason. Combinatorial relations and chromatic
graphs, Cauad.J. Math., 7. 1955. I 7.
S. Ui'DiTNiF.Mi, "Disconnected-colorings of Graphs". Combinatorial structures,
Gordon and Breach. New York. 1970, 163 167.
Z. IIi.driIn, An application of the Ramsey theorem to the topologicul product,
Bull. Acad. Polon. Sc, 14. 1966. 25 26.
J. G. KALnri fiscii, R. G. Stanton. On the Maximal Triangle l-'rce Edge Chro-
Chromatic Graphs in Three Colors, J. Comh. Theory. 5. 1968, 9 20.
M. Las Veronas. Sur un problcmc d'Erdos ct IIajnal. J. Comb. Theory, 11, 1971,
39 44.
L. Lovasz, On Decomposition of Graphs. Stadia Sc. Math, liuugarica, I, 1966,
237-238.
L. LovAsz. On Chromatic Number of Finite Set-systems. Acta Mathematlca Ac.
Sc. Hung., 19, 1968,59 67.
I. Lovasz, "Graphs and Set Systems", lieitriige zur Grapheutheorle (II. Sachs,
II. J. Voss, II. Walthcr, cd.). Tcubncr, 1968. 99 106.
T. S. Mot/kin, "Colorings, Cocolorings and Determinant Terms". Theory of
Graphs, Rome I. C. C. (l\ Roscnslichl. cd.). Dunod. Paris, 1967, 253 254.
J. Mycillski, Sur Iccoloriugc des graphes. Colloq. Math., 3, 1955, 161 162.
J. NiiSi-.rftiL. On A-chromatic Graphs (Russian). Comm. Math. Univ. Carol., 7,
1966, 3 10.
J. NeSmril. P. IIi:ll, "^-societies with Given Semi-group". Combinatorial
Structures and their Applications, Gordon and Breach, New York. 1970.
301-392.
F. P. Ramsfy, On a Problem of Formal Logic, Proc. London Math. Soc, 30,
1930,264 286.
A. Rosa. "On the Chromatic Number of Steincr Triple Systems". Combinatorial
structures, Gordon and Breach. New York, 1970, 369 371.
H. Sachs. "Finite graphs". Recent Progress in Combinatorics (Tutte, ed.), Aca-
Academic Press. New York, 1969, 175 183.
II. Sachs. M. Sc-iiXuni.t, "Obcr die Konstruklion von Graphcn mil gewissen
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A. J. HorrMAN, J. B. Kruskal, "Integral Boundary Points of Convex Poly-
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M. Las Veronas, A Decomposition Theorem for Hypcrgraphs, Minieo., J. H. P.,
1970.
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21, 1970,443-446.
L. LovAsz, Normal Hypcrgraphs and the Perfect Graph Conjecture. Discrete Math.,
2, 1972, 253-267.
L. Mirsky, II. Pfrfect, The Distribution of Positive Elements in Doubly Sto-
chasiic Matrices. J. London Math. Soc., 40, 1965, 689- 698.
D. 1% WwtRA. Equitable color itions of graphs, R.I.R.O., R-3. 1971, 3-8.
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Index of Definitions
(English, French, German)
The following index gives the page numbers of the terms defined in this
text and their French and German equivalents. Synonyms arc given in
italics.
For more specific definitions, the reader is referred to the following excellent
glossaries:
J. W. Essain. M. E. Fisher. Some Basic Definitions in Graph Theory,
Reviews of Modern Physics, 42. 271-88, April. 1970 (English).
R. Pellet, Initiation d la TMorie des Greplies, Entreprise modcrne d'Edition,
Paris, 1966 (French).
abnorhant. externally stable.
303
absorption number, fHG),
303
adjacent
to an arc, 5
to a Tuce. 18
to a ict, 4
to a vertex, 4
in hypergraphi, 389
adjoint graph. 38
adjoint mutrU, 142
alternating
chain, 12)
sequence. 275
sequence in hypergruph*.
414
anti-node, 30
anti-symmetric, 7
urboreaoence. rooted trtt. 33
arc, tllrettttl edge, 3
arc (low, 76
articulation set
eul-srl. separating net, 8
articulation vertex
cut-vertex, separating v rtex, 9
associated matrix, 9
auocialcd number, r(x), 61
bulanccd hypergraph, 450
abiorbant, exterleurement
stable
nombre d'abiorptlon, dc
siabiliit extern
adjacent
adjoint
matrice adjointe
ulternte
anti-nitud
antlnymitrlque
arborcicence
arc
flim
enicmble d'articulatlon
point d'urtlculation
matrice is&ocite
nombre ■ssocM
tquilibri
fiusserlKh slubil
■wick Swbililtls uhl
adjazent
Bogenmltieniiraph
Adjunktmatrix
aliernierend
Antiknotenpunkt
antitymmctrlwh •
BUtchel, Setsbaum
Rogen
I-Ium
Artikulation, Mrfalltnde
Menge
Zerfullunituknolenpunkl
Adja/enx Matrix
bciiteordncie Zalil
Im Cilclchgcwicht
523

524
INDEX OF DEFINITIONS
bull
ofcocyclei, 15
of cycle*, 15
of ■ mitrold, 493
blcolourlng, 325
bipartite graph, 7
network, 83
block. 175
Boolean complement, 420
Boolean product, 420
Boolean sum, 420
boundary, 18
branch, track, 30
cictul, Hutlml Trtt, 175
cipicily. 76
Cirlctlun product
of 1-graph*, 314
of simple graph*, 377
Cartesian sum
of 1-graph*, 314
of umple graph*, 376
cell, 285
centre, 61
chain
In ■ graph, 7
In ■ hypergraph, 391
chord, diagonal, lit
chromatic number
of a graph, y(G). 325
of ■ hypergraph *(//), 428
chromatic index, 4@), 248
chromatic polynomial. 352
circuit, dlrtcled circuit. 8
circuit (in a mutroid). 476
clique
of ■ graph, 7
of ■ hypergraph, 432
cocircuit. 13
cocycle. eohoumhry, 13
cocyclomatlc number. MO), 15
fc-colourable, 325
colouring
of edges. 248
good—of edge*. 440
of vcrticc*, 325
for ■ hypergraph. 428
comparability graph. 363
complementary graph
of ■ l-graph, 189
or ■ timple graph. 288
complete graph. 7
complete bipartite graph, 7
component, connected
component
of ■ graph, 8
of tt hypergraph. 391
composition product, 74
conformul. with a faithful troph
rrprestnlatlon, 3%
bailt
blcoloratlon
biparll, blchromatiqut
bloc
complementation boolcenne
prodult boolcenne
somme boolcenne
(rontiere
branche
CICIUS
capacltc
produit cartttien
somme cartcilenne
cellule
centre
chulne
corde, dlagomile
nombre chromatique
indice chromatique. elasst
CnrunnitH/UC
polynome
circuit
mlgme
clique, n-clique
cocircuit
cocycle
nombre cocyclomatique
fc-chromutique
coloration
graphe de comparability
graphe complcmentaire
complet
biparti-complet
compotante
produit direct
conformc
Basil
ZweirArbung
pair, blchromallsch
Block
Boolc-Komplement
Boole-Produkt
Roole-Summe
Rand
AM. Zweig
Kaktu*
KipiiJtll
kartetitchet Produkt
kartealtche Summe
Zclle
Zentrum
Kantenfolge
Sehne
chromatitche Zahl
chromutiiche KlasM
chromatlKhei Polynom
geiichlouene Bogenfolge.
7irkiiil
fclFKUII
Stigmui
Clique
Cozirkuit
Cozyklus
cozyklomatitche Zahl
fc-chromatiKh
Rrbung
Verglclchngraph
komplementarer Graph.
Kom piemen t
volUtandlg
vollmindig-paur
Komponente
Kompmitionsproduki
konform

INDEX OF DEFINITIONS
525
conjugate sequences, 103
connected. 8
A-connecicd, 164
connectivity, 164
contour, 18
contraction, 31
cotree. 26
covering, cover, 129
covering number, 489
^■covering, d-covtr, 419
critical. 285, 338, 424
crossing number
cubic gripli, regular of degree. 3
cut, arc cutset, 82
cycle
of ■ graph, 8
of • hypergraph, 391
cyclomalic number, i>(G), 15
dark vertex, 1 $5
deflclency, 417
degree, valency
of ■ graph. 6
of a hypergraph. 429
density, ui(G'), maximum
cardinality of a clique
descendant
diameter, J(G>, 66
«*(C/>. 7J
digital turn. 317
directed distance, 61
dominance number, /3*(G),
domination numb r, 305
dual. 390
edge, line, 4
of a hypergraph, 389
ct!ge-conn«clivlty. 182
fc-edge-connccied, 1H2
elementary chain, 8
elementary circuit, 8
elementary cocyclc, 13
elementary contraction, 350
elementary cycle, 12
elementary path, 8
equitable colouring, 463
eulerian, 228
face, region, 18
factor, 230
Ferrer's diagram, 103
flow, 76
forest, 24
four-colour conjecture, 280
functional, 37
genus, «(C)
girth, length of the longest cycle
graph, directed graph, 3
conjugue'
connexe
fc-connexe
connectivity
contour
contraction, riirtclssement
coarbre
recouvrcment
nombre de recouvrement
rf-recouvrcment
critique
nombre de crolscment
cublque
coupe
cycle
nombre cyclomatlque
ipals
deficlence
degr«
denslti
descendant
diametre
somme digitale
teart
nombre de ttabiliti externe
dual
artte
connexil*
fc-arttc-connexe
chalne eitmentaire
circuit dimentaire
cocycle ilimentalre
contraction ilimenuire
cycle (SltSmenliire
chemin iMmcntaire
coloration Equitable
eultrlen
face
facteur, scmi-facteur
dlugramme de Ferrers
Hot
fortl
conjecture det quatre
couleurs
ronciionnd, univoque
genre
calibre
genre
konjugiert
zuiammenhtngend
fc-xunammenhangend
Zunammenhangszahl
Kontur
Kontraktlon
Co-Bium
Cberdeckung. Decke
Cberdeckungzahl
(/•Cberdeckung
krltltch
0 berkreuiunguahl,
Cberschneidungtzahl
kublsch
Schnitt
Zyklut
zyklomatltche Zahl
dick
Fehlmenge
Crad, Valenz
Dlchtc
Nachkomme
Durchmesser
digitals Summe
orientierte Entfernung
tussere Stabilitatscahl
dual
Kante
Kantenzuiammenhang
fc-fach kantenzuummen-
lilngend
Weg
Elementarzirkuit
elementarer Cozyklus
elementare Kontraktion
Elementarzyklui, Krelt
elementare Bahn
gleichwertige Firbung
Eulerscher
Fllche
quadratitcher Faktor
Ferrers-Diagramm
Strom
Wald
Vlerfarben-Vermutung
funktlonal
Art
Taillenweite
Graph (gerlchtel, mil oder
ohne mchrfuche BOgen,
mil oder ohne Schlingen)

526
INDEX OF DEFINITIONS
p-groph, directed graph with
IrutltipilCliy p, J
Grundy function, 312
homiltonian, 186
Hamilton-connected, 217
Helly property, 397
hypergraph, stl system. 389
inaccessible vertex, 155
incident, 6
incidence mutrix, 389
independent set
of cycles, cocyclcs, 15
in matroids, 476
Injectlve. 36
Inner deml-dcgree. deml-dtgrtt
Inward, Inwofettct, 6
interval griph, 371
isolated vertex, 4
Isomorphlc, 411
Isthmus, 175
kernel, 307
length, 7
light vertex, 155
loop, 3
matching, packing
of a graph, 122
of a hypergraph, 414
c-matching, 150
matrold, 476
maximal set, 10
maximum set, 10
minimum set, 10
minimally-connected graph, 30
^-minimal graph, 442
minimum set, 10
mixed vertex, 155
multigraph, 5
multiple edge, 108
multiplicity, 6
neighbour, 4
network, 77
Mm, 319
node, junction point, 30
normal hypergraph, 458
order
of a graph, 3
of a hypergruph, 389
outer dcmi-dcgrcc, deml-degret
outward, oul-valence, 6
partial
— subgruph, 7
— hypergruph. 390
— subhypergruph, 390
path, directed path, 8
pendant vertex, 25
perfect graph, 360
perfect matching, linear factor
122
p-graphe
fonctlon de Grundy
hamiltonien
Hamilton-connect*
proprlcte de Helly
hypergriphe
Inaccessible
incident
matricc d'lncldence
ensemble Independent
injectif
deml-degri interleur
griphe d'intervilles
Isole
isomorphe
Isthme
noyau
longueur
An
boucle
couplage
c-couplage
matrolde
maximal
maximum
mlnlmll
connexe-minimul
4-mlnlmal
minimum
mixte
multigraphe
arete multiple
multiplicity
voitin
reseau
jeu de Nim
noeud, carrefour
normal
ordre
demi-detri exterleur
particl
chcmln
sommet pendant
perfait
couplage porfuit
p-Graph (lerlchtct)
Grundy-Funktlon
Hamiltonscher
Hamilton-zuummenhlngcnd
Helly Elgenschaft
Hypergraph
uncrrclchbir
Inzident
InzJdenzmatrlx
independentc Menge
Injcktiv
Eingangsgrad, Elngangs-
lntervallgraph
Isollert
Uomorph
Iithmus, Brtlcke
Kem
Llngc
dunn
Schllnge
Paarung, Anhlufung
c-Paarung
Matrold
maximal
maximal kardinal
minimal
minimal zuiammenhtngend
4-mlnlmal
minimal kardinal
gemischt
Multigraph, ungertchteter
f iianh
urupn
Mchrfachkante
Multiplizltu
Nnchbar
Nell, Austauschnetz
Nlm-Splcl
Mehrfachknotenpunkt,
Verzwelgung
normal
Ordnung
Ausgangsgrad, Ausgangi-
valenz
Tcll-
Bogenfolge
Endknotcnpunkt
perfekt
1 inearfaktor

INDFX OF DEFINITIONS
527
piece. 329
planar. 17
potential, 91
predecessor, 4
product
of I-graphs, 314
of simple graphs, 377
pseudo-cycle, 8
pseudo-symmetric, 239
quadrilateral, elemtntary cyclt
of length, 4
quasi-strongly connected graph,
32
radius, 61
Ramsey number, 436
rank, 390
regular, homogeneous, 6 .
representative iruph, llne-grJlpli,
400
root, 32
rosace, 30
saturated, 122
fc-scciion. 390
semi-bipartite, 141
scmi-functlonal, 37
sieve, 422
simple chain, R
simple circuit, ft
simple graph, ordinary graph,
5
simple hypergruph, 389
simple path, 8
sink. 77
source, 77
spanning tree, 26
stability number, Indtptndence
number, 272
of a hypergraph, 428
stable, Independent, 9
strong stability number, 448
strong ^-colouring, 448
strongly connected component
stronx toniponent, 28
strongly connected, 28
stochastic function. 469
Miburaph. imluctil suhgraplt. 7
subnypcrgraph, 390
successor, 3
sum
of cycles. 12
ofcocyclc*. 12
support, 469
symmetric graph, 6
temion, potential difference, 91
piece
plunuire
potentiel
predceesscur
produit normal
ptcudo-cyclc
pseudo-symftrique
quadrilitere
quasl-fortcment connexe
rayon
nombre de Ramtey
rang
regullcr, homogenc
graphc representatif
raclne
rosace
saturj
fc-section
seml-bipurti
scmi-fonctionncl, semi-
univoque
grille
chulne simple
cireuit simple
graphe simple
hypergraphc simple
chemin simple
sortie
entree
arbre maximal
nombre de stabilite, stablllte
interne
stable, Intirleurement stable
nombre de stability forte
9-coiorotion forte
composante fortement
connexe
fortement connexe
fonction stochastique
soui-grnphe
sous-hypergraphe
successeur
somme
support
graphe sytnetrique
tension, dilKrence de
potentiel
died. Bluit
planar
Potential
Vorgangcr
normalei Produkt
Pseudo-Zyklus
pseudosymmetrhch
Vierkren
quasi-stark zummmen-
hangend
Radius
Kamsey-Zahl
Rungfunktion
regulir
entsprechender Graph
Wurzcl
Roiette
geiatiigt
fc-Schnitt
scmi-paur
semi-funktional
Citter
Kanteruug
einfacher Zirkuit
(schlichter) Graph
(schlichtcr) Hypergraph
Bahn
Ausging, Senke
Eingang, Quelle
GerQtt
innere Stabllitfltszahl
inncrlich stabii
»tarke StubilitllHuihl
sturke 9-h'flrbung
sturk-zuiammenhlkngende
Komponente
stark zusimmenhilngend
stovhastische Funktlon
Untergraph
Unterhypergraph
Nochfolger
Summo
StiltM
symmetrischer Graph
Spannung

528
INDEX Of DEFINITIONS
transportation network, 77
transversal
fora graph. 133
for ■ hypcrgraph. 420
transversal number, 420
tree, 24
triangle, cycle of length, 3
triangle hypcrgraph. 440
triangulated, 368
Turin number. 434
uniform hypergraph, It-graph,
lOfl
unimodular
hypcrgraph. 465
matrix, 468
vertex, point, 3
thickness
topological graph. 18
topolouical dual, 21
transfer, 152
transitive graph, 310
riseau de transport
trnnsversul
nombre de tranivcmlite
arbre
triangle
hypergraphe des triangles
triangult
nombre de Turin
unlforme
unimodulalre
sommet, point
tpaliseur
gruphe topologlque
dual
transfer!
trannitif
Transport neu
transversal
Transversalltauzahl
Buum
Drelkrels
Dreikrels-Hypcrgraph
trianguliert
Turan-Zahl
gleichfUrmig
unimodular
Knotenpunkt
Dicke
topologischer Graph
dual
Austautch
tramitiv