Chromatic graph theory

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 Theory explores 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 many distance-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.

0 The Origin of Graph Colorings	p. 1
1 Introduction to Graphs	p. 27
1.1 Fundamental Terminology	p. 27
1.2 Connected Graphs	p. 30
1.3 Distance in Graphs	p. 33
1.4 Isomorphic Graphs	p. 37
1.5 Common Graphs and Graph Operations	p. 39
1.6 Multigraphs and Digraphs	p. 44
Exercises for Chapter 1	p. 47
2 Trees and Connectivity	p. 53
2.1 Cut-vertices, Bridges, and Blocks	p. 53
2.2 Trees	p. 56
2.3 Connectivity and Edge-Connectivity	p. 59
2.4 Menger's Theorem	p. 63
Exercises for Chapter 2	p. 67
3 Eulerian and Hamiltonian Graphs	p. 71
3.1 Eulerian Graphs	p. 71
3.2 de Bruijn Digraphs	p. 76
3.3 Hamiltonian Graphs	p. 79
Exercises for Chapter 3	p. 87
4 Matchings and Factorization	p. 91
4.1 Matchings	p. 91
4.2 Independence in Graphs	p. 98
4.3 Factors and Factorization	p. 100
Exercises for Chapter 4	p. 106
5 Graph Embeddings	p. 109
5.1 Planar Graphs and the Euler Identity	p. 109
5.2 Hamiltonian Planar Graphs	p. 118
5.3 Planarity Versus Nonplanarity	p. 120
5.4 Embedding Graphs on Surfaces	p. 131
5.5 The Graph Minor Theorem	p. 139
Exercises for Chapter 5	p. 141
6 Introduction to Vertex Colorings	p. 147
6.1 The Chromatic Number of a Graph	p. 147
6.2 Applications of Colorings	p. 153
6.3 Perfect Graphs	p. 160
Exercises for Chapter 6	p. 170
7 Bounds for the Chromatic Number	p. 175
7.1 Color-Critical Graphs	p. 175
7.2 Upper Bounds and Greedy Colorings	p. 179
7.3 Upper Bounds and Oriented Graphs	p. 189
7.4 The Chromatic Number of Cartesian Products	p. 195
Exercises for Chapter 7	p. 200
8 Coloring Graphs on Surfaces	p. 205
8.1 The Four Color Problem	p. 205
8.2 The Conjectures of Hajos and Hadwiger	p. 208
8.3 Chromatic Polynomials	p. 211
8.4 The Heawood Map-Coloring Problem	p. 217
Exercises for Chapter 8	p. 219
9 Restricted Vertex Colorings	p. 223
9.1 Uniquely Colorable Graphs	p. 223
9.2 List Colorings	p. 230
9.3 Precoloring Extensions of Graphs	p. 240
Exercises for Chapter 9	p. 245
10 Edge Colorings of Graphs	p. 249
10.1 The Chromatic Index and Vizing's Theorem	p. 249
10.2 Class One and Class Two Graphs	p. 255
10.3 Tait Colorings	p. 262
10.4 Nowhere-Zero Flows	p. 269
10.5 List Edge Colorings	p. 279
10.6 Total Colorings of Graphs	p. 282
Exercises for Chapter 10	p. 284
11 Monochromatic and Rainbow Colorings	p. 289
11.1 Ramsey Numbers	p. 289
11.2 Turan's Theorem	p. 296
11.3 Rainbow Ramsey Numbers	p. 299
11.4 Rainbow Numbers of Graphs	p. 306
11.5 Rainbow-Connected Graphs	p. 314
11.6 The Road Coloring Problem	p. 320
Exercises for Chapter 11	p. 324
12 Complete Colorings	p. 329
12.1 The Achromatic Number of a Graph	p. 329
12.2 Graph Homomorphisms	p. 335
12.3 The Grundy Number of a Graph	p. 349
Exercises for Chapter 12	p. 356
13 Distinguishing Colorings	p. 359
13.1 Edge-Distinguishing Vertex Colorings	p. 359
13.2 Vertex-Distinguishing Edge Colorings	p. 370
13.3 Vertex-Distinguishing Vertex Colorings	p. 379
13.4 Neighbor-Distinguishing Edge Colorings	p. 385
Exercises for Chapter 13	p. 391
14 Colorings, Distance, and Domination	p. 397
14.1 T-Colorings	p. 397
14.2 L(2, 1)-Colorings	p. 403
14.3 Radio Colorings	p. 410
14.4 Hamiltonian Colorings	p. 417
14.5 Domination and Colorings	p. 425
14.6 Epilogue	p. 434
Exercises for Chapter 14	p. 434
Appendix 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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