Chromatic Graph Theory

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 GraphsFundamental TerminologyConnected 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 GraphsFactors and Factorization Graph Embeddings Planar Graphs and the Euler Identity Hamiltonian Planar GraphsPlanarity 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 ColoringsUniquely 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 ColoringsTotal 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 GraphGraph 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.

Gary Chartrand, Ping Zhang