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Chromatic Graph Theory

Gary Chartrand and Ping Zhang
Chapman & Hall/CRC
Publication Date: 
Number of Pages: 
Discrete Mathematics and Its Applications 53
[Reviewed by
Miklós Bóna
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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
Connectivity and Edge-Connectivity
Menger’s Theorem
Eulerian and Hamiltonian Graphs
Eulerian Graphs
de Bruijn Digraphs
Hamiltonian Graphs
Matchings and Factorization
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
L(2, 1)-Colorings
Radio Colorings
Hamiltonian Colorings
Domination and Colorings
Appendix: Study Projects
General References
Index (Names and Mathematical Terms) 
List of Symbols 
Exercises appear at the end of each chapter.