Pearls in Graph Theory: A Comprehensive Introduction

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Overview

A stimulating view of mathematics that appeals to students as well as teachers, this undergraduate-level text is written in an informal style that does not sacrifice depth or challenge. Based on twenty years of teaching by the leading researcher in graph theory, it offers a solid foundation on the subject. This revised and augmented edition features new exercises, simplifications, and other improvements suggested by classroom users and reviewers. The first three chapters -- on basic graph theory, colorings of graphs, and circuits and cycles -- are basic to the understanding of the rest of the text. Chapters 6, 8, 9, and 10 -- on labeling graphs, drawings of graphs, measurements of closeness to planarity, and graphs on surfaces -- are appropriate for a mathematically intensive course. Chapter 7, on applications and algorithms, is of particular importance to computer science students. Chapters 4 and 5, on extremal problems and counting, are recommended if time is available.
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Product Details

  • ISBN-13: 9780123285522
  • Publisher: Elsevier Science & Technology Books
  • Publication date: 7/1/1990
  • Pages: 246

Table of Contents

Foreword to the Revised Edition vii
Foreword ix
Chapter 1. Basic Graph Theory
1.1 Graphs and Degrees of Vertices 1
1.2 Subgraphs, Isomorphic Graphs 13
1.3 Trees 17
Chapter 2. Colorings of Graphs
2.1 Vertex Colorings 23
2.2 Edge Colorings 31
2.3 Decompositions and Hamilton Cycles 35
2.4 More Decomposition 43
Chapter 3. Circuits and Cycles
3.1 Eulerian Circuits 49
3.2 The Oberwolfach Problem 58
3.3 Infinite Lattice Graphs 65
Chapter 4. Extremal Problems
4.1 A Theorem of Turan 70
4.2 Cages 77
4.3 Ramsey Theory 81
Chapter 5. Counting
5.1 Counting 1-Factors 87
5.2 Cayley's Spanning Tree Formula 94
5.3 More Spanning Trees 100
Chapter 6. Labeling Graphs
6.1 Magic Graphs and Graceful Trees 104
6.2 Conservative Graphs 114
Chapter 7. Applications and Algorithms
7.1 Spanning Tree Algorithms 125
7.2 Matchings in Graphs, Scheduling Problems 133
7.3 Binary Trees and Prefix Codes 140
Chapter 8. Drawings of Graphs
8.1 Planar Graphs 149
8.2 The Four Color Theorem 156
8.3 The Five Color Theorem 164
8.4 Graphs and Geometry 168
Chapter 9. Measurements of Closeness to Planarity
9.1 Crossing Number 179
9.2 Thickness and Splitting Number 190
9.3 Heawood's Empire Problem 197
Chapter 10. Graphs on Surfaces
10.1 Rotations of Graphs 208
10.2 Planar Graphs Revisited 217
10.3 The Genus of a Graph 225
References 241
Index 245
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