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El. knyga: Chromatic Graph Theory

3.67/5 (6 ratings by Goodreads)

Ping Zhang (Western Michigan University, Kalamazoo, Michigan, USA), Gary Chartrand

Formatas: 525 pages
Serija: Textbooks in Mathematics
Išleidimo metai: 28-Nov-2019
Leidėjas: CRC Press
ISBN-13: 9780429798283

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Formatas: 525 pages
Serija: Textbooks in Mathematics
Išleidimo metai: 28-Nov-2019
Leidėjas: CRC Press
ISBN-13: 9780429798283

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With Chromatic Graph Theory, Second Edition, the authors present 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. Readers will see that the authors accomplished the primary goal of this textbook, which is to introduce graph theory with a coloring theme and to look at graph colorings in various ways.

The textbook also 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.

Features of the Second Edition:

The book can be used for a first course in graph theory as well as a graduate course

The primary topic in the book is graph coloring

The book begins with an introduction to graph theory so assumes no previous course

The authors are the most widely-published team on graph theory

Many new examples and exercises enhance the new edition

Preface To The Second Edition

List of Symbols

xix

0 The Origin of Graph Colorings

(26)

1 Introduction to Graphs

(26)

1.1 Fundamental Terminology

(3)

1.2 Connected Graphs

(4)

1.3 Distance in Graphs

(3)

1.4 Isomorphic Graphs

(2)

1.5 Common Graphs and Graph Operations

(6)

1.6 Multigraphs and Digraphs

(8)

Exercises for
Chapter 1

(5)

2 Trees and Connectivity

(18)

2.1 Cut-Vertices, Bridges, and Blocks

(3)

2.2 Trees

(4)

2.3 Connectivity and Edge-Connectivity

(3)

2.4 Menger's Theorem

(8)

Exercises for
Chapter 2

(4)

3 Eulerian and Hamiltonian Graphs

(20)

3.1 Eulerian Graphs

3.2 De Bruijn Digraphs

(3)

3.3 Hamiltonian Graphs

(12)

Exercises for
Chapter 3

(4)

4 Matchings and Factorization

(18)

4.1 Matchings and Independence

(8)

4.2 Factorization and Decomposition

(10)

Exercises for
Chapter 4

105

(4)

5 Graph Embeddings

109

(38)

5.1 Planar Graphs and the Euler Identity

109

(9)

5.2 Hamiltonian Planar Graphs

118

(2)

5.3 Planarity versus Nonplanarity

120

(9)

5.4 Outerplanar Graphs

129

(2)

5.5 Embedding Graphs on Surfaces

131

(8)

5.6 The Graph Minor Theorem

139

(8)

Exercises for
Chapter 5

141

(6)

6 Introduction to Vertex Colorings

147

(28)

6.1 The Chromatic Number of a Graph

147

(6)

6.2 Applications of Colorings

153

(7)

6.3 Perfect Graphs

160

(15)

Exercises for
Chapter 6

170

(5)

7 Bounds for the Chromatic Number

175

(30)

7.1 Color-Critical Graphs

175

(4)

7.2 Upper Bounds and Greedy Colorings

179

(10)

7.3 Upper Bounds and Oriented Graphs

189

(6)

7.4 The Chromatic Number of Cartesian Products

195

(10)

Exercises for
Chapter 7

200

(5)

8 Coloring Graphs on Surfaces

205

(18)

8.1 The Four Color Problem

205

(3)

8.2 The Conjectures of Hajos and Hadwiger

208

(3)

8.3 Chromatic Polynomials

211

(6)

8.4 The Heawood Map-Coloring Problem

217

(6)

Exercises for
Chapter 8

220

(3)

9 Restricted Vertex Colorings

223

(26)

9.1 Uniquely Colorable Graphs

223

(7)

9.2 List Colorings

230

(10)

9.3 Precoloring Extensions of Graphs

240

(9)

Exercises for
Chapter 9

245

(4)

10 Edge Colorings

249

(40)

10.1 The Chromatic Index and Vizing's Theorem

249

(6)

10.2 Class One and Class Two Graphs

255

(7)

10.3 Tait Colorings

262

(7)

10.4 Nowhere-Zero Flows

269

(10)

10.5 List Edge Colorings

279

(3)

10.6 Total Colorings of Graphs

282

(7)

Exercises for
Chapter 10

284

(5)

11 Ramsey Theory

289

(26)

11.1 Ramsey Numbers

289

(8)

11.2 Bipartite Ramsey Numbers

297

(8)

11.3 s-Bipartite Ramsey Numbers

305

(10)

Exercises for
Chapter 11

311

(4)

12 Monochromatic Ramsey Theory

315

(28)

12.1 Monochromatic Ramsey Numbers

315

(4)

12.2 Monochromatic-Bichromatic Ramsey Numbers

319

(3)

12.3 Proper Ramsey Numbers

322

(5)

12.4 Rainbow Ramsey Numbers

327

(6)

12.5 Gallai-Ramsey Numbers

333

(10)

Exercises for
Chapter 12

340

(3)

13 Color Connection

343

(32)

13.1 Rainbow Connection

344

(8)

13.2 Proper Connection

352

(11)

13.3 Rainbow Disconnection

363

(12)

Exercises for
Chapter 13

372

(3)

14 Distance and Colorings

375

(30)

14.1 T-Colorings

376

(5)

14.2 L(2, l)-Colorings

381

(6)

14.3 Radio Colorings

387

(7)

14.4 Hamiltonian Colorings

394

(11)

Exercises for
Chapter 14

402

(3)

15 Domination and Colorings

405

(18)

15.1 Domination Parameters

405

(3)

15.2 Stratified Domination

408

(6)

15.3 Domination Based on AT3-Colorings

414

(4)

15.4 Stratified Domination by Multiple Graphs

418

(5)

Exercises for
Chapter 15

420

(3)

16 Induced Colorings

423

(38)

16.1 Majestic Colorings

423

(9)

16.2 Royal and Regal Colorings

432

(16)

16.3 Rainbow Mean Colorings

448

(13)

Exercises for
Chapter 16

456

(5)

17 The Four Color Theorem Revisited

461

(14)

17.1 Zonal Labelings of Planar Graphs

461

(3)

17.2 Zonal Labelings and Edge Colorings

464

(11)

Exercises for
Chapter 17

472

(3)

Bibliography

475

(13)

Index (Names and Mathematical Terms)

488

Gary Chartrand is a professor emeritus of mathematics at Western Michigan University.

Ping Zhang is a professor of mathematics at Western Michigan University.

The two have authored or co-authored many textbooks in mathematics and numerous research articles in graph theory. The authors publish several books on graph theory, including the best-selling Graphs and Diagraphs, Sixth Edition, CRC Press, the most widely-used introductory text for course in graph theory.

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