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基本説明
Covers all the basic material in full detail, and adds one or two deeper results (again with detailed proofs) to illustrate the more advanced methods.
Full Description
The third edition of this standard textbook of modern graph theory has been carefully revised, updated, and substantially extended. Covering all its major recent developments it can be used both as a reliable textbook for an introductory course and as a graduate text: on each topic it covers all the basic material in full detail, and adds one or two deeper results (again with detailed proofs) to illustrate the more advanced methods of that field. From the reviews of the first two editions (1997, 2000): "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". "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". "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".
Contents
Preface 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 Packing and covering 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: Extremal Graph Theory 7.1 Subgraphs* 7.2 Minors(*) 7.3 Hadwiger's conjecture* 7.4 Szemeredi'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 The topological end space 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 Simple 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* 1 1.4 Threshold functions and second moments Exercises Notes 12: Minors, Trees and WQO 12.1 Well-quasi-ordering* 12.2 The graph minor theorem for trees* 12.3 Tree-decompositions 12.4 Tree-width and forbidden minors 12.5 The graph minor theorem(*) Exercises Notes A. Infinite sets B. Surfaces Hints for all the exercises Index Symbol index * Sections marked by an asterisk are recommended for a first course. Of sections marked (*), the beginning is recommended for a first course.
