6th Edition

# Graphs & Digraphs

640 Pages 343 B/W Illustrations
by Chapman & Hall

Graphs & Digraphs masterfully employs student-friendly exposition, clear proofs, abundant examples, and numerous exercises to provide an essential understanding of the concepts, theorems, history, and applications of graph theory.

Fully updated and thoughtfully reorganized to make reading and locating material easier for instructors and students, the Sixth Edition of this bestselling, classroom-tested text:

• Adds more than 160 new exercises
• Presents many new concepts, theorems, and examples
• Includes recent major contributions to long-standing conjectures such as the Hamiltonian Factorization Conjecture, 1-Factorization Conjecture, and Alspach’s Conjecture on graph decompositions
• Supplies a proof of the perfect graph theorem
• Features a revised chapter on the probabilistic method in graph theory with many results integrated throughout the text

At the end of the book are indices and lists of mathematicians’ names, terms, symbols, and useful references. There is also a section giving hints and solutions to all odd-numbered exercises. A complete solutions manual is available with qualifying course adoption.

Graphs & Digraphs, Sixth Edition remains the consummate text for an advanced undergraduate level or introductory graduate level course or two-semester sequence on graph theory, exploring the subject’s fascinating history while covering a host of interesting problems and diverse applications.

Introduction
Graphs
The Degree of a Vertex
Isomorphic Graphs
Regular Graphs
Bipartite Graphs
Operations on Graphs
Degree Sequences
Multigraphs
Exercises for Chapter 1

Connected Graphs and Distance
Connected Graphs
Distance in Graphs
Exercises for Chapter 2

Trees
Nonseparable Graphs
Introduction to Trees
Spanning Trees
The Minimum Spanning Tree Problem
Exercises for Chapter 3

Connectivity
Connectivity and Edge-Connectivity
Theorems of Menger and Whitney
Exercises for Chapter 4

Eulerian Graphs
The Königsberg Bridge Problem
Eulerian Circuits and Trails
Exercises for Chapter 5

Hamiltonian Graphs
Hamilton's Icosian Game
Sufficient Conditions for Hamiltonicity
Toughness of Graphs
Highly Hamiltonian Graphs
Powers of Graphs and Line Graphs
Exercises for Chapter 6

Digraphs
Introduction to Digraphs
Strong Digraphs
Eulerian and Hamiltonian Digraphs
Tournaments
Kings in Tournaments
Hamiltonian Tournaments
Exercises for Chapter 7

Flows in Networks
Networks
The Max-Flow Min-Cut Theorem
Menger Theorems for Digraphs
Exercises for Chapter 8

Automorphisms and Reconstruction
The Automorphism Group of a Graph
Cayley Color Graphs
The Reconstruction Problem
Exercises for Chapter 9

Planar Graphs
The Euler Identity
Maximal Planar Graphs
Characterizations of Planar Graphs
Hamiltonian Planar Graphs
Exercises for Chapter 10

Nonplanar Graphs
The Crossing Number of a Graph
The Genus of a Graph
The Graph Minor Theorem
Exercises for Chapter 11

Matchings, Independence and Domination
Matchings
1-Factors
Independence and Covers
Domination
Exercises for Chapter 12

Factorization and Decomposition
Factorization
Decomposition
Cycle Decomposition
Graceful Graphs
Exercises for Chapter 13

Vertex Colorings
The Chromatic Number of a Graph
Color-Critical Graphs
Bounds for the Chromatic Number
Exercises for Chapter 14

Perfect Graphs and List Colorings
Perfect Graphs
The Perfect and Strong Perfect Graph Theorems
List Colorings
Exercises for Chapter 15

Map Colorings
The Four Color Problem
Colorings of Planar Graphs
List Colorings of Planar Graphs
The Conjectures of Hajós and Hadwiger
Chromatic Polynomials
The Heawood Map-Coloring Problem
Exercises for Chapter 16

Edge Colorings
The Chromatic Index of a Graph
Class One and Class Two Graphs
Tait Colorings
Exercises for Chapter 17

Nowhere-Zero Flows, List Edge Colorings
Nowhere-Zero Flows
List Edge Colorings
Total Colorings
Exercises for Chapter 18

Extremal Graph Theory
Turán's Theorem
Extremal Subgraphs
Cages
Exercises for Chapter 19

Ramsey Theory
Classical Ramsey Numbers
More General Ramsey Numbers
Exercises for Chapter 20

The Probabilistic Method
The Probabilistic Method
Random Graphs
Exercises for Chapter 21

Hints and Solutions to Odd-Numbered Exercises

Bibliography

Supplemental References

Index of Names

Index of Mathematical Terms

List of Symbols

### Biography

Gary Chartrand is a professor emeritus of mathematics at Western Michigan University, Kalamazoo, Michigan, USA. Linda Lesniak, a professor emeritus of mathematics from Drew University, Madison, New Jersey, USA, is currently a visiting mathematician at Western Michigan University, Kalamazoo, Michigan, USA. Ping Zhang is a professor of mathematics at Western Michigan University, Kalamazoo, Michigan, USA. All three have authored or coauthored many textbooks in mathematics and numerous research articles in graph theory.

Praise for the Previous Edition

"Now in its fifth edition, its success as a textbook is indicative of its quality and its clarity of presentation. … The authors also describe the fascinating history behind some of the key problems in graph theory and, to a lesser extent, their applications. This book describes the key concepts you need to get started in graph theory. … It provides all you might need to know about graph embeddings and graph colorings. Moreover, it analyzes many other topics that more general discrete mathematics monographs do not always cover, such as network flows, minimum cuts, matchings, factorization, decomposition, and even extremal graph theory. … This thorough textbook includes hundreds of exercises at the end of each section. Hints and solutions for odd-numbered exercises are included in the appendix, making it especially suitable for self-learning."
—Fernando Berzal, Computing Reviews, September 2011

"As with the earlier editions, the current text emphasizes clear exposition, well-written proofs, and many original and innovative exercises of varying difficulty and challenge. … The fifth edition continues and extends these fine traditions."
—Arthur T. White, Zentralblatt MATH 1211