GRAPH THEORY WITH APPLICATIONS

1 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 Bondy and Muny 1976. First published in Great Britain 1976 by The Macmillan Press Ltd. First published in the 1976 by Elsevier Science Publishing Co., Inc. 52 Vanderbilt Avenue, New York, 10017. Fifth Printing, 1982. Sole Distributor in the : Elsevier Science Publishing Co., Inc. Library of Congress Cataloging in Publication Data Bondy, John Adrian. GRAPH THEORY with APPLICATIONS . Bibliography: p. lncludes index.

2 1. GRAPH THEORY . 1. Murty, , joint author. II. Title. 1979 511 '.5 75-29826. ISBN 0.:444-19451-7. 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 America To 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, Chv tal, Tutte and Vizing.

3 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.

4 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.

5 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. Lov sz 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 the Preface vii manuscript and valuable suggestions, and to the ubiquitous G.

6 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 vi 1 GRAPHS AND SUBGRAPHS. Graphs and Simple Graphs . 1. GRAPH Isomorphism 4. The Incidence and Adjacency Matrices 7. Subgraphs 8. Vertex Degrees 10.

7 Paths and Connection 12. Cycles 14. APPLICATIONS The Shortest Path Problem. 15. Sperner's Lemma. 21. 2 TREES. Trees 25. Cut Edges and Bonds .. 27. Cut Vertices. 31. Cayley's Formula . 32. APPLICATIONS The Connector Problem 36. 3 CONNECTIVITY. Connectivity . 42. Blocks . 44. APPLICATIONS Construction of Reliable Communication Networks 47. 4 EULER TOURS AND HAMILTON CYCLES. Euler Tours . 51. Hamilton Cycles . 53. APPLICATIONS Postman Problem 62. The Travelling Salesman Problem 65. Contents ix 5 MATCHINGS. 5 .1 Matchings 70. 5 .2 Matchings and Coverings in Bipartite Graphs 72. Perfect Matchings.

8 76. APPLICATIONS The Personnel Assignment Problem 80. The Optimal Assignment Problem 86..6 EDGE COLOURINGS. Edge Chromatic Number 91. Vizing's Theorem . 93. APPLICATIONS 63 The Timetabling Problem 96. 7 INDEPENDENT SETS AND CLIQUES. Independent Sets .. 101. Ramsey's Theorem 103. 7 .3 Turan 's Theorem .. 109. APPLICATIONS Schur's Theorem . 112. A Geometry Problem . 113. 8 VERTEX COLOURINGS. Chroniatic Number ..117. Brooks' Theorem .. 122. Haj6s' Conjecture. 123. Chromatic Polynomial~. 125. Girth and Chromatic Number 129. APPLICATIONS A Storage Problem 131. 9 PLANAR GRAPHS. Plane and Planar Graphs 135.

9 Dual Graphs . 139. Euler's Formula .. 143. Bridges .. 145. Kuratowski's Theorem .. 151. The Five-Colour Theorem and the Four-Colour Conjecture 156. Nonhamiltonian Planar Graphs . 160. APPLICATIONS 9 .8 A Planarity Algorithm .. 163. X Contents 10 DIRECTED GRAPHS. Directed Graphs . 171. Directed Paths 173. Directed Cycles 176. APPLICATIONS A Job Sequencing Problem. 179. Designing an Efficient Computer Drum 181. Making a Road System One-Way 182. Ranking the Participants in a Tournament. 185. 11 NETWORKS. Flows 191. Cuts 194. The Max-Flow Min-Cut Theorem 196. APPLICATIONS Menger's Theorems 203.

10 Feasible Flows 206. 12 THE CYCLE SPACE AND BOND SPACE. Circulations and Potential Differences. 212. The Number of Spanning Trees . 218. APPLICATIONS Perfect Squares . 220. Appendix I Hints to Starred Exercises 227. Appendix II Four Graphs and a Table of their Properties . 232. Appendix III Sorne Interesting Graphs. 234. Appendix IV Unsolved Problems. 246. Appendix V Suggestions for Further Reading 254. Glossary of Symbols 257. Index 261. 1 Graphs and Subgraphs GRAPHS AND SIMPLE GRAPHS. Many real-world situations can conveniently be described by means of a diagram consisting of a set of points together with lines joining certain pairs of these points.