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Publisher:

Chapman & Hall/CRC

Publication Date:

2009

Number of Pages:

483

Format:

Hardcover

Series:

Discrete Mathematics and Its Applications 53

Price:

99.95

ISBN:

9781584888000

Category:

Monograph

[Reviewed by , on ]

Miklós Bóna

12/18/2008

This is a well-conceived and well-written book. The authors decided to write an unusally long but very welcome introductory part. This part takes up the first five chapters, and it makes the book self-contained, since it covers basic notions from the definitions of graph and degrees to trees and connectivity, to Eulerian and Hamiltonian graphs, matchings, and planar graphs. Chromatic graph theory comes only after that, in the remaining nine chapters.

These chapters (6–14) are exclusively about graph colorings, roughly equally divided between vertex colorings and edge colorings. Beside the classic topics, such as Ramsey theory, the Turán theorem, and the chromatic number of a graph, we can also learn about less well-known notions, such as the Grundy number of a graph, Rainbow Ramsey numbers, and Radio colorings. The balance between old and new is well struck.

The fact that more than one-third of the book is about graphs in general makes the book suitable not just for a course in chromatic graph theory, but for a course in graph theory in general. This is a major feat — the authors managed to write about a special topic without losing the chance for a wide audience. Whether the course taught from the book is advanced undergraduate or first-year graduate will depend on the school.

The book is written in a reader-friendly style, and there is a sufficient number of exercises at the end of each chapter. My only critical remark is that none of these exercises come with solutions, or even hints. This may not bother professors teaching a course from the book, but will surely upset students, especially those learning on their own, or taking a reading course.

Miklós Bóna is Associate Professor of Mathematics at the University of Florida.

**The Origin of Graph Colorings** **Introduction to Graphs**

Fundamental Terminology

Connected Graphs

Distance in Graphs

Isomorphic Graphs

Common Graphs and Graph Operations

Multigraphs and Digraphs**Trees and Connectivity**

Cut Vertices, Bridges, and Blocks

Trees

Connectivity and Edge-Connectivity

Menger’s Theorem**Eulerian and Hamiltonian Graphs**

Eulerian Graphs

de Bruijn Digraphs

Hamiltonian Graphs**Matchings and Factorization**

Matchings

Independence in Graphs

Factors and Factorization**Graph Embeddings**

Planar Graphs and the Euler Identity

Hamiltonian Planar Graphs

Planarity versus Nonplanarity

Embedding Graphs on Surfaces

The Graph Minor Theorem**Introduction to Vertex Colorings**

The Chromatic Number of a Graph

Applications of Colorings

Perfect Graphs**Bounds for the Chromatic Number**

Color-Critical Graphs

Upper Bounds and Greedy Colorings

Upper Bounds and Oriented Graphs

The Chromatic Number of Cartesian Products**Coloring Graphs on Surfaces**

The Four Color Problem

The Conjectures of Hajós and Hadwiger

Chromatic Polynomials

The Heawood Map-Coloring Problem**Restricted Vertex Colorings**

Uniquely Colorable Graphs

List Colorings

Precoloring Extensions of Graphs**Edge Colorings of Graphs**

The Chromatic Index and Vizing’s Theorem

Class One and Class Two Graphs

Tait Colorings

Nowhere-Zero Flows

List Edge Colorings

Total Colorings of Graphs**Monochromatic and Rainbow Colorings**

Ramsey Numbers

Turán’s Theorem

Rainbow Ramsey Numbers

Rainbow Numbers of Graphs

Rainbow-Connected Graphs

The Road Coloring Problem**Complete Colorings**

The Achromatic Number of a Graph

Graph Homomorphisms

The Grundy Number of a Graph**Distinguishing Colorings**

Edge-Distinguishing Vertex Colorings

Vertex-Distinguishing Edge Colorings

Vertex-Distinguishing Vertex Colorings

Neighbor-Distinguishing Edge Colorings**Colorings, Distance, and Domination***T*-Colorings*L*(2, 1)-Colorings

Radio Colorings

Hamiltonian Colorings

Domination and Colorings

Epilogue**Appendix: Study Projects****General References****Bibliography** **Index (Names and Mathematical Terms)** **List of Symbols** *Exercises appear at the end of each chapter.*

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