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GRAPH THEORY WITH APPLICATIONS
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GRAPH THEORY
WITH APPLICATIONS
J. A. Bondy and U. S. R. Murty
Department of Combina tories and Optimization,
University of Waterloo,
Ontario,
Canada
NORfH-HOLLAND
New York • Amsterdam • Oxford
@J.A. Bondy and U.S.R. Muny 1976First published in Great Britain 1976 by
The Macmillan Press Ltd.
First published in the U.S.A. 1976 by
Elsevier Science Publishing
Co., Inc.
52 Vanderbilt Avenue, New York, N.Y. 10017
Fifth Printing, 1982.
Sole Distributor
in the U.S.A:Elsevier Science Publishing
Co., Inc.
Library
of Congress Cataloging in Publication DataBondy, John Adrian.
Graph theory with applications.
Bibliography: p.
lncludes index.1. Graph theory.
QA166.B67 1979
ISBN 0.:444-19451-7
1. Murty, U.S.R., joint author. II. Title.
511 '.5 75-29826
AU rights reserved. No part of this publication may be reproduced or transmitted, in any form or by any means, without permission.Printed
in the United States of AmericaTo our parents
Preface
This book is intended as an introduction to graph theory. Our aim bas been to present what we consider to be the basic material, together with a wide variety of applications, both to other branches of mathematics and to real-world problems. Included are simple new proofs of theorems of Brooks, Chvâtal, Tutte and Vizing. The applications have been carefully selected, and are treated in some depth. We have chosen to omit ail so-called 'applications' that employ just the language of graphs and no theory. The applications appearing at the end of each chapter actually make use of theory developed earlier in the same chapter. We have also stressed the importance of efficient methods of solving problems. Several good al gorithms are included and their efficiencies are analysed. We do not, however, go into the computer iinplementation of these algorithms.The exercises at the
end of each section are of varying difficulty. The harder ones are starred (*) and, for these, hints are provided in appendix I. ln some exercises, new . definitions · are introduced. The reader is recom mended to acquaint himself with these definitions. Other exercises, whose numbers are indicated by bold type, are used in subsequent sections; these should ail be attempted. Appendix II consists of a table in which basic properties of four graphs are listed. When new definitions are introduced,· the reader may find it
helpful to check bis understanding by referring to this table. Appendix III includes a selection of interesting graphs with special properties. These may prove to be useful in testing new conjectures. In appendix IV, we collect together a number of unsolved problems, some known to be very difficult, and others more hopeful. Suggestions for further reading are given in appendix V.Many people have contributed, either directly
or indirectly, to this book.We are particularly indebted to C. Berge and D.
J. ~-Welsh for introducing
us to graph theory, to G. A. Dirac,J. Edmonds, L. Lovâsz and W. T. Tutte,
whose works have influenced oui-treatment of the subject, to V. Chungphaisan and C. St. J. A. Nash-Williams for their careful reading of thePreface vii
manuscript and valuable suggestions, and to the ubiquitous G. O. M. for his kindness and constant encouragement.We also wish to thank
S. B. Maurer, P. J. O'Halloran, C. Thomassen,
B. Toft and our colleagues at the University of Waterloo for many helpful comments, and the National Research Council of Canada for its financial support. Finally, we would like to express our appreciation to Joan Selwood for her excellent typing and Diana Rajnovich for her beautiful artwork.J. A. Bondy
U.S. R. Murty
Contents
Pre/ace
1 GRAPHS AND SUBGRAPHS
1.1 Graphs and Simple Graphs .
1.2 Graph Isomorphism
1.3The Incidence and Adjacency Matrices
1.4Subgraphs
1.5 Vertex Degrees
1.6Paths and Connection
1.7 Cycles
Applications
1.8 The Shortest Path Problem.
1.9 Sperner's Lemma.
2 TREES
2.1 Trees
2.2Cut Edges and Bonds
2.3 Cut Vertices.
2.4Cayley's Formula .
Applications
2.5 The Connector Problem
3 CONNECTIVITY
3 .1 Connectivity .
3.2 Blocks .
Applications
3.3 Construction of Reliable Communication Networks
4 EULER TOURS AND HAMILTON CYCLES
4.1 Euler Tours .
4.2 Hamilton Cycles .
Applications
4.3 The.Chinese Postman Problem
4.4 The Travelling Salesman Problem
vi 1 4 7 8 10 12 14 15 2125
27
31
32
36
42
44
47
51
53
62
65
Contents
5 MATCHINGS
5 .1 Matchings
5 .2 Matchings and Coverings in Bipartite Graphs
5.3 Perfect Matchings .
Applications
5.4 The Personnel Assignment Problem ·
5.5 The Optimal Assignment Problem
. 6 EDGE COLOURINGS6.1 Edge Chromatic Number
6.2 Vizing's Theorem .
Applications
63 The Timetabling Problem
7 INDEPENDENT SETS AND CLIQUES
7.1 Independent Sets .
7.2 Ramsey's Theorem
7 .3Turan 's Theorem .
Applications
7.4 Schur's Theorem .
7.5 A Geometry Problem .
8 VERTEX COLOURINGS
8.1 Chroniatic Number
8.2 Brooks' Theorem .
8.3 Haj6s'
· Conjecture.
8.4 Chromatic
Polynomial~.
8.5 Girth and Chromatic Number
Applications
8.6 A Storage Problem
9 PLANAR GRAPHS
ix 7072
76
80
86
91
93
96
. 101 103
. 109 112
113
. .117 . 122 123
125
129
131
9.1 Plane and Planar Graphs 135
9.2 Dual Graphs . 139
9.3 Euler's Formula . . 143
9.4 Bridges . . 145
9.5 Kuratowski's Theorem . . 151
9.6 The Five-Colour Theorem and the Four-Colour Conjecture 156
9.7 Nonhamiltonian Planar Graphs . 160
Applications
9 .8 A Planarity Algorithm . . 163
X10 DIRECTED GRAPHS
10.1 Directed Graphs .
10.2Directed Paths
10.3Directed Cycles
Applications
10.4 A Job Sequencing Problem.
10.5 Designing an Efficient Computer Drum
10.6 Making a Road System One-Way
10.7 Ranking the Participants in a Tournament.
11 NETWORKS
11.1 Flows
11.2 Cuts11.3 The Max-Flow Min-Cut Theorem
Applications
11.4 Menger's Theorems
11.5 Feasible Flows
12 THE CYCLE SPACE AND BOND SPACE
12.1 Circulations and Potential Differences.
12.2The Number of Spanning Trees .
Applications
12.3 Perfect Squares .
Appendix I
Appendix II
Appendix III
Appendix IV
Appendix V
Hints to Starred Exercises
Four Graphs and a Table
of their Properties .Sorne Interesting Graphs.
Unsolved Problems.
Suggestions
for Further ReadingGlossary
of SymbolsIndex Contents
171173
176
179
181
182
185
191
194
196
203
206
212
218
220
227
232
234
246
254
257
261