Chromatic Graph Theory

Chromatic Graph Theory

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Features

  • 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

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 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.

Table of Contents

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.


Editorial Reviews

… 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

 
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