Chromatic Graph Theory

by Chartrand; Gary

• ISBN13: 9781584888000
• ISBN10: 1584888008
• Format: Hardcover
• Copyright: 2008-09-22
• Publisher: Chapman & Hall/

Summary

Beginning with the origin of the four color problem in 1852, the field of graph colorings has developed into one of the most popular areas of graph theory. Introducing graph theory with a coloring theme, Chromatic Graph Theoryexplores connections between major topics in graph theory and graph colorings as well as emerging topics.This self-contained book first presents various fundamentals of graph theory that lie outside of graph colorings, including basic terminology and results, trees and connectivity, Eulerian and Hamiltonian graphs, matchings and factorizations, and graph embeddings. The remainder of the text deals exclusively with graph colorings. It covers vertex colorings and bounds for the chromatic number, vertex colorings of graphs embedded on surfaces, and a variety of restricted vertex colorings. The authors also describe edge colorings, monochromatic and rainbow edge colorings, complete vertex colorings, several distinguishing vertex and edge colorings, and manydistance-related vertex colorings.With historical, applied, and algorithmic discussions, this text offers a solid introduction to one of the most popular areas of graph theory.

Table of Contents

The Origin of Graph Colorings	p. 1
Introduction to Graphs	p. 27
Fundamental Terminology	p. 27
Connected Graphs	p. 30
Distance in Graphs	p. 33
Isomorphic Graphs	p. 37
Common Graphs and Graph Operations	p. 39
Multigraphs and Digraphs	p. 44
Exercises for Chapter 1	p. 47
Trees and Connectivity	p. 53
Cut-vertices, Bridges, and Blocks	p. 53
Trees	p. 56
Connectivity and Edge-Connectivity	p. 59
Menger's Theorem	p. 63
Exercises for Chapter 2	p. 67
Eulerian and Hamiltonian Graphs	p. 71
Eulerian Graphs	p. 71
de Bruijn Digraphs	p. 76
Hamiltonian Graphs	p. 79
Exercises for Chapter 3	p. 87
Matchings and Factorization	p. 91
Matchings	p. 91
Independence in Graphs	p. 98
Factors and Factorization	p. 100
Exercises for Chapter 4	p. 106
Graph Embeddings	p. 109
Planar Graphs and the Euler Identity	p. 109
Hamiltonian Planar Graphs	p. 118
Planarity Versus Nonplanarity	p. 120
Embedding Graphs on Surfaces	p. 131
The Graph Minor Theorem	p. 139
Exercises for Chapter 5	p. 141
Introduction to Vertex Colorings	p. 147
The Chromatic Number of a Graph	p. 147
Applications of Colorings	p. 153
Perfect Graphs	p. 160
Exercises for Chapter 6	p. 170
Bounds for the Chromatic Number	p. 175
Color-Critical Graphs	p. 175
Upper Bounds and Greedy Colorings	p. 179
Upper Bounds and Oriented Graphs	p. 189
The Chromatic Number of Cartesian Products	p. 195
Exercises for Chapter 7	p. 200
Coloring Graphs on Surfaces	p. 205
The Four Color Problem	p. 205
The Conjectures of Hajos and Hadwiger	p. 208
Chromatic Polynomials	p. 211
The Heawood Map-Coloring Problem	p. 217
Exercises for Chapter 8	p. 219
Restricted Vertex Colorings	p. 223
Uniquely Colorable Graphs	p. 223
List Colorings	p. 230
Precoloring Extensions of Graphs	p. 240
Exercises for Chapter 9	p. 245
Edge Colorings of Graphs	p. 249
The Chromatic Index and Vizing's Theorem	p. 249
Class One and Class Two Graphs	p. 255
Tait Colorings	p. 262
Nowhere-Zero Flows	p. 269
List Edge Colorings	p. 279
Total Colorings of Graphs	p. 282
Exercises for Chapter 10	p. 284
Monochromatic and Rainbow Colorings	p. 289
Ramsey Numbers	p. 289
Turan's Theorem	p. 296
Rainbow Ramsey Numbers	p. 299
Rainbow Numbers of Graphs	p. 306
Rainbow-Connected Graphs	p. 314
The Road Coloring Problem	p. 320
Exercises for Chapter 11	p. 324
Complete Colorings	p. 329
The Achromatic Number of a Graph	p. 329
Graph Homomorphisms	p. 335
The Grundy Number of a Graph	p. 349
Exercises for Chapter 12	p. 356
Distinguishing Colorings	p. 359
Edge-Distinguishing Vertex Colorings	p. 359
Vertex-Distinguishing Edge Colorings	p. 370
Vertex-Distinguishing Vertex Colorings	p. 379
Neighbor-Distinguishing Edge Colorings	p. 385
Exercises for Chapter 13	p. 391
Colorings, Distance, and Domination	p. 397
T-Colorings	p. 397
L(2, 1)-Colorings	p. 403
Radio Colorings	p. 410
Hamiltonian Colorings	p. 417
Domination and Colorings	p. 425
Epilogue	p. 434
Exercises for Chapter 14	p. 434
Study Projects	p. 439
General References	p. 446
Bibliography	p. 453
Index (Names and Mathematical Terms)	p. 465
List of Symbols	p. 480
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