eBook

- Presents the fundamentals of graph theory, including trees, Hamiltonian graphs, factorizations, and more
- Covers key topics in graph theory, such as Ramsey numbers and domination
- Explores emerging areas in graph coloring, including list colorings, rainbow colorings, distance colorings related to the channel assignment problem, and vertex/edge distinguishing colorings
- Contains carefully explained proofs and examples, along with many exercises of varying levels of difficulty at the end of each chapter
- Offers suggestions for study projects in the appendix

*Solutions manual available for qualifying instructors*

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.

**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.*

… The book is written in a student-friendly style with carefully explained proofs and examples and contains many exercises of varying difficulty. … The book is intended for standard courses in graph theory, reading courses and seminars on graph colourings, and as a reference book for individuals interested in graphs colourings.

—*Zentralblatt MATH* 1169

… well-conceived and well-written book … written in a reader-friendly style, and there is a sufficient number of exercises at the end of each chapter.

—Miklós Bóna, University of Florida, *MAA Online*, January 2009