Graphs & Digraphs: (Textbooks in Mathematics)

Graphs & Digraphs, Seventh Edition masterfully employs student-friendly exposition, clear proofs, abundant examples, and numerous exercises to provide an essential understanding of the concepts, theorems, history, and applications of graph theory. This classic text, widely popular among students and instructors alike for decades, is thoroughly streamlined in this new, seventh edition, to present a text consistent with contemporary expectations. Changes and updates to this edition include: A rewrite of four chapters from the ground up Streamlining by over a third for efficient, comprehensive coverage of graph theory Flexible structure with foundational Chapters 1–6 and customizable topics in Chapters 7–11 Incorporation of the latest developments in fundamental graph theory Statements of recent groundbreaking discoveries, even if proofs are beyond scope Completely reorganized chapters on traversability, connectivity, coloring, and extremal graph theory to reflect recent developments The text remains the consummate choice for an advanced undergraduate level or introductory graduate-level course exploring the subject’s fascinating history, while covering a host of interesting problems and diverse applications. Our major objective is to introduce and treat graph theory as the beautiful area of mathematics we have always found it to be. We have striven to produce a reader-friendly, carefully written book that emphasizes the mathematical theory of graphs, in all their forms. While a certain amount of mathematical maturity, including a solid understanding of proof, is required to appreciate the material, with a small number of exceptions this is the only pre-requisite. In addition, owing to the exhilarating pace of progress in the field, there have been countless developments in fundamental graph theory ever since the previous edition, and many of these discoveries have been incorporated into the book. Of course, some of the proofs of these results are beyond the scope of the book, in which cases we have only included their statements. In other cases, however, these new results have led us to completely reorganize our presentation. Two examples are the chapters on coloring and extremal graph theory.

Table of Contents:
1 Graphs 1.1 Fundamentals 1.2 Isomorphism 1.3 Families of graphs 1.4 Operations on graphs 1.5 Degree sequences 1.6 Path and cycles 1.7 Connected graphs and distance 1.8 Trees and forests 1.9 Multigraphs and pseudographs 2 Digraphs 2.1 Fundamentals 2.2 Strongly connected digraphs 2.3 Tournaments 2.4 Score sequences 3 Traversability 3.1 Eulerian graphs and digraphs 3.2 Hamiltonian graphs 3.3 Hamiltonian digraphs 3.4 Highly hamiltonian graphs 3.5 Graph powers 4 Connectivity 4.1 Cut-vertices, bridges, and blocks 4.2 Vertex connectivity 4.3 Edge-connectivity 4.4 Menger's theorem 5 Planarity 5.1 Euler's formula 5.2 Characterizations of planarity 5.3 Hamiltonian planar graphs 5.4 The crossing number of a graph 6 Coloring 6.1 Vertex coloring 6.2 Edge coloring 6.3 Critical and perfect graphs 6.4 Maps and planar graphs 7 Flows 7.1 Networks 7.2 Max-flow min-cut theorem 7.3 Menger's theorems for digraphs 7.4 A connection to coloring 8 Factors and covers 8.1 Matchings and 1-factors 8.2 Independence and covers 8.3 Domination 8.4 Factorizations and decompositions 8.5 Labelings of graphs 9 Extremal graph theory 9.1 Avoiding a complete graph 9.2 Containing cycles and trees 9.3 Ramsey theory 9.4 Cages and Moore graphs 10 Embeddings 10.1 The genus of a graph 10.2 2-Cell embeddings of graphs 10.3 The maximum genus of a graph 10.4 The graph minor theorem 11 Graphs and algebra 11.1 Graphs and matrices 11.2 The automorphism group 11.3 Cayley color graphs 11.4 The reconstruction problem