Graph Theory (5th ed.)

Main subject category: • Graph theoryMathematics Subject Classification (2010): • 05-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to combinatorics • 05Cxx Graph theoryThis standard textbook of modern graph theory, now in its fifth edition, combines the authority of a classic with the engaging freshness of style that is the hallmark of active mathematics. It covers the core material of the subject with concise yet reliably complete proofs, while offering glimpses of more advanced methods in each field by one or two deeper results, again with proofs given in full detail.The book can be used as a reliable text for an introductory course, as a graduate text, and for self-study. Title Page Preface First edition Second edition Third edition Fourth edition Fifth edition Contents 1. The Basics 1.1 Graphs 1.2 The degree of a vertex 1.3 Paths and cycles 1.4 Connectivity 1.5 Trees and forests 1.6 Bipartite graphs 1.7 Contraction and minors 1.8 Euler tours 1.9 Some linear algebra 1.10 Other notions of graphs Exercises Notes 2. Matching, Covering and Packing 2.1 Matching in bipartite graphs 2.2 Matching in general graphs 2.3 The Erdös-Pósa theorem 2.4 Tree packing and arboricity 2.5 Path covers Exercises Notes 3. Connectivity 3.1 2-Connected graphs and subgraphs 3.2 The structure of 3-connected graphs 3.3 Menger’s theorem 3.4 Mader’s theorem 3.5 Linking pairs of vertices Exercises Notes 4. Planar Graphs 4.1 Topological prerequisites 4.2 Plane graphs 4.3 Drawings 4.4 Planar graphs: Kuratowski’s theorem 4.5 Algebraic planarity criteria 4.6 Plane duality Exercises Notes 5. Colouring 5.1 Colouring maps and planar graphs 5.2 Colouring vertices 5.3 Colouring edges 5.4 List colouring 5.5 Perfect graphs Exercises Notes 6. Flows 6.1 Circulations 6.2 Flows in networks 6.3 Group-valued flows 6.4 k-Flows for small k 6.5 Flow-colouring duality 6.6 Tutte’s flow conjectures Exercises Notes 7. ExtremalGraph Theory 7.1 Subgraphs 7.2 Minors 7.3 Hadwiger’s conjecture 7.4 Szemerédi’s regularity lemma 7.5 Applying the regularity lemma Exercises Notes 8. Infinite Graphs 8.1 Basic notions, facts and techniques 8.2 Paths, trees, and ends 8.3 Homogeneous and universal graphs 8.4 Connectivity and matching 8.5 Recursive structures 8.6 Graphs with ends: the complete picture 8.7 The topological cycle space 8.8 Infinite graphs as limits of finite ones Exercises Notes 9. Ramsey Theory for Graphs 9.1 Ramsey’s original theorems 9.2 Ramsey numbers 9.3 Induced Ramsey theorems 9.4 Ramsey properties and connectivity Exercises Notes 10. Hamilton Cycles 10.1 Sufficient conditions 10.2 Hamilton cycles and degree sequences 10.3 Hamilton cycles in the square of a graph Exercises Notes 11. Random Graphs 11.1 The notion of a random graph 11.2 The probabilistic method 11.3 Properties of almost all graphs 11.4 Threshold functions and second moments Exercises Notes 12. Graph Minors 12.1 Well-quasi-ordering 12.2 The graph minor theorem for trees 12.3 Tree-decompositions 12.4 Tree-width 12.5 Tangles 12.6 Tree-decompositions and forbidden minors 12.7 The graph minor theorem Exercises Notes A. Infinite sets B. Surfaces Hints for all theExercises Hints for Chapter 1 Hints for Chapter 2 Hints for Chapter 3 Hints for Chapter 4 Hints for Chapter 5 Hints for Chapter 6 Hints for Chapter 7 Hints for Chapter 8 Hints for Chapter 9 Hints for Chapter 10 Hints for Chapter 11 Hints for Chapter 12 Index Symbol Index This standard textbook of modern graph theory, now in its fifth edition, combines the authority of a classic with the engaging freshness of style that is the hallmark of active mathematics. It covers the core material of the subject with concise yet reliably complete proofs, while offering glimpses of more advanced methods in each field by one or two deeper results, again with proofs given in full detail. The book can be used as a reliable text for an introductory course, as a graduate text, and for self-study. From the reviews: "This outstanding book cannot be substituted with any other book on the present textbook market. It has every chance of becoming the standard textbook for graph theory."Acta Scientiarum Mathematiciarum "Deep, clear, wonderful. This is a serious book about the heart of graph theory. It has depth and integrity. "Persi Diaconis & Ron Graham, SIAM Review "The book has received a very enthusiastic reception, which it amply deserves. A masterly elucidation of modern graph theory." Bulletin of the Institute of Combinatorics and its Applications "Succeeds dramatically ... a hell of a good book." MAA Reviews "A highlight of the book is what is by far the best account in print of the Seymour-Robertson theory of graph minors." Mathematika " ... like listening to someone explain mathematics." Bulletin of the AMS This standard textbook of modern graph theory, now in its fifth edition, combines the authority of a classic with the engaging freshness of style that is the hallmark of active mathematics. It covers the core material of the subject with concise yet reliably complete proofs, while offering glimpses of more advanced methods in each field by one or two deeper results, again with proofs given in full detail. The book can be used as a reliable text for an introductory course, as a graduate text, and for self-study. From the reviews: “This outstanding book cannot be substituted with any other book on the present textbook market. It has every chance of becoming the standard textbook for graph theory.” Acta Scientiarum Mathematiciarum “Deep, clear, wonderful. This is a serious book about the heart of graph theory. It has depth and integrity.” Persi Diaconis & Ron Graham, SIAM Review “The book hasreceived a very enthusiastic reception, which it amply deserves. A masterly elucidation of modern graph theory.” Bulletin of the Institute of Combinatorics and its Applications “Succeeds dramatically... a hell of a good book.” MAA Reviews “A highlight of the book is what is by far the best account in print of the Seymour-Robertson theory of graph minors.” Mathematika “... like listening to someone explain mathematics.” Bulletin of the AMS