Table Of ContentTheory and Application
of Graphs
Network Theory and Applications
Valurne 10
Managing Editors:
Ding-ZhuDu
University ofMinnesota, U.S.A.
Cauligi Raghavendra
UniversityofSouthern Califorina, U.SA.
Theory and Application
of Graphs
by
Junming Xu
Department of Mathematics
U niversity of Science and Technology of China
Hefei, Anhui, China
Springer Science+ Business Media, LLC
Library of Congress Cataloging-in-Publication
CIP info or:
Title: Theory and Application ofGraphs
Author: Xu
ISBN 978-1-4613-4670-8 ISBN 978-1-4419-8698-6 (e Book)
DOI 10.1007/978-1-4419-8698-6
Copyright© 2003 by Springer Science+Business Media New York
Originally published by Kluwer Academic Publishers in 2003
Softcover reprint of the bardeover 1st edition 2003
All rights reserved. No partoftbis publication may be reproduced, stored in a retrieval system
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Permissions forbooks published in the USA: permj ssj ons@wkap CQJD
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Printed on acid-Jree paper.
Contents
Preface vii
1 Basic Concepts of Graphs 1
1.1 Graph and Graphical Presentation 2
1.2 Graph Isomorphism . . . . 7
1.3 Vertex Degrees ....... . 14
1.4 Subgraphs and Operations . . 19
1.5 Walks, Paths and Connection 25
1.6 Distauces and Diameters . 32
1. 7 Circuits and Cycles . 38
1.8 Eulerian Graphs . . . . . 47
1.9 Hamiltonian Graphs ... 54
1.10 Matrix Presentation of Graphs 62
Applications
1.11 Exponents of Primitive Matrices 70
2 Trees and Graphie Spaces 79
2.1 Trees and Spanning Trees 80
2.2 Vector Spaces of Graphs . 88
2.3 Enumeration of Spanning Trees 100
Applications
2.4 The Minimum Connector Problem 107
2.5 The Shortest Path Problem .... 113
2.6 The Electrical Network Equations 122
3 Plane Graphs and Planar Graphs 127
3.1 PlaneGraphsand Euler's Formula 128
3.2 Kuratowski's Theorem 137
3.3 Dual Graphs . . . . . 143
Applications
Contents
VI
3.4 Regular Polyhedra . . . . . 148
3.5 Layout of Printed Circuits . 151
4 Flows and Connectivity 159
4.1 Network Flows .. 160
4.2 Menger's Theorem 165
4.3 Connectivity . . . 174
Applications
4.4 Design of Transport Schemes ..... 183
4.5 Design of Optimal Transport Schemes 191
4.6 The Chinese Postman Problem ... 197
4. 7 Construction of Squared Rectangles 204
5 Matchings and Independent Sets 211
5.1 Matchings 212
5.2 Independent Sets 227
Applications
5.3 The Personnel Assignment Problem 232
5.4 The Optimal Assignment Problem 241
5.5 The Travelling Salesman Problem . 250
6 Coloring Theory 257
6.1 Vertex Colorings 258
6.2 Edge Colarings . 265
Applications
6.3 The Four-Color Problem . 271
7 Graphs and Groups 279
7.1 Group Presentation of Graphs . 280
7.2 Transitive Graphs ...... . 285
7.3 Graphie Presentation of Groups . 294
Applications 301
7.4 Design of Interconnection Networks . 301
Bibliography 309
List of Symbols 323
Index 327
Preface
In the spectrum of mathematics, graph theory which studies a mathe
matical structure on a set of elements with a binary relation, as a recognized
discipline, is a relative newcomer. In recent three decades the exciting and
rapidly growing area of the subject abounds with new mathematical devel
opments and significant applications to real-world problems. More and more
colleges and universities have made it a required course for the senior or the
beginning postgraduate students who are majoring in mathematics, computer
science, electronics, scientific management and others. This book provides
an introduction to graph theory for these students.
The richness of theory and the wideness of applications make it impossi
ble to include all topics in graph theory in a textbook for one semester. All
materials presented in this book, however, I believe, are the most classical,
fundamental, interesting and important. The method we deal with the mate
rials is to particularly lay stress on digraphs, regarding undirected graphs as
their special cases. My own experience from teaching out of the subject more
than ten years at University of Science and Technology of China (USTC)
shows that this treatment makes hardly the course di:fficult, but much more
accords with the essence and the development trend of the subject.
The book consists of seven chapters. Excepting that the first chapter
contains the most basic concepts and results, each chapter presents one spe
cial topic, including trees and graphic spaces, plane and planar graphs, flows
and connectivity, matchings and independent sets, coloring theory, graphs
and groups. These topics are treated in some depth, both theoretical and ap
plied, with some suggestions for further reading. Every effort will be made to
strengthen the mutual connections among these topics, with an aim to make
the materials more systematic and cohesive. All theorems will be stated
clearly, tagether with full and concise proofs, some of them are new. A
number of examples and more than 350 figures are given to help the reader
VIII Preface
to understand the given materials. To explore the mathematical nature of
graph theory better, we will specially stress the equivalence of some classi
cal results, such as the max-flow min-cut theorem of Ford and Fulkerson,
Menger's theorem, Hall's theorem, Tutte's theorem and König's theorem.
Throughout this book the reader will see that graph theory has closed
connection with other branches of mathematics, including linear algebra,
matrix theory, group theory, combinatorics, combinatorial optimization and
operation research, and wide applications to other subjects, including Com
puter science, electronics, scientific management and so on. The applications
carefully selected are arranged in the latter section(s) of each chapter. The
aim of such arrangements is to conveniently choose these materials for some
readers according to their interesting and available periods.
Exercises at the end of each section, more than 500, from routine practice
to challenging, are Supplements to the text. Some of them are very important
results in graph theory. It is advisable for the reader to be familiar with the
new definitions introduced in the exercises since they are useful for further
study. The reader is also advised to do the exercises as many as he (o r she)
can. The harder ones are indicated by bold type.
The style of writing and of presentation of this book have be, to a great
extent, influenced by Graph Theory with Applications, a popular textbook
written by J. A. Bondy and U. S. R. Murty whom I am grateful to, from
which some typical materials have been directly selected in this book.
The book is developed from the textforasenior and first-year postgrad
uate course in one semester at USTC. I thank the participants of the course
for their great interest and stimulating comments. I would like to thank
Teaching Affairs Division, Graduate School and Department of Mathematics
at USTC for their support and encouragement.
Many people have contributed, either directly or indirectly, to this book. I
avail myself of this opportunity to particularly express my heartfelt gratitude
to Qiao Li, Feng Tian, Yanpei Liu, Genghua Fan, Yongchuan Chen, Dingzhu
Du and Shenggui Zhang for their continuous help and valuable suggestions.
Finally, I would like to express my appreciation to my son, Keli Xu, for his
very concrete help, and my wife, Jingxia Qiu, for her support, understanding
and love, without which this work would have been impossible.
Jun-Ming Xu
([email protected])
December 2002
Chapter 1
Basic Concepts of Graphs
In many real-world situations, it is particularly convenient to describe the
specified relationship between pairs of certain given objects by means of a
diagram, in which points presents the objects and (directed or undirected)
Iines the relationship between pairs of the objects. For example, a national
traffic map describes a condition of the communication lines among cities
in the country, where the points represent cities and the lines represent the
highways or the railways joining pairs of cities. Notice that in such diagrams
one is mainly interested in whether or not two given points are joined by
a Iine; the manner in which they are joined is immaterial. A mathematical
abstraction of situations of this type gives rise to the concept of a graph.
In fact, a graph provides the natural structures from which to construct
mathematical models that are appropriate to almost all fields of scientific
(natural and social) inquiry. The underlying subject of study in these fields
is some set of "objects" and one or more "relations" between the objects.
In this chapter, we will introduce basic concepts of a graph used in the
remaining parts of the book, including several special graphs, subgraphs of
various types, walk, path, cycle, diameter, connection, Euler circuit, Rarnil
ton cycle, adjacency and incidence matrices, as weil as the basic results closely
related these concepts. At the end of this chapter we will present an appli
cation of graphs to matrix theory.
It should, for the beginner specially, be worth noting that most graph
theorists use personalized terminology in their books, papers and lectures.
Even the meaning of the word "graph" varies with different authors. We will
adopt the most standard terminology and notation extensively used by most
authors, with a subject index and a Iist of symbols in the end of the book.
Description:In the spectrum of mathematics, graph theory which studies a mathe matical structure on a set of elements with a binary relation, as a recognized discipline, is a relative newcomer. In recent three decades the exciting and rapidly growing area of the subject abounds with new mathematical devel o
