紀伊國屋書店

基本説明

A highly successful textbook has been carefully revised and updated, and includes a new chapter on infinite graphs. Can be used both as a reliable textbook for an introductory course and as a graduate text.

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

Preface1: 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 algebra1.10 Other notions of graphsExercisesNotes2: Matching, Covering and Packing2.1 Matching in bipartite graphs*2.2 Matching in general graphs(*)2.3 Packing and covering2.4 Tree-packing and arboricity 2.5 Path covers ExercisesNotes3: 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: Flows6.1 Circulations(*)6.2 Flows in networks* 6.3 Group-valued flows6.4 k-Flows for small k6.5 Flow-colouring duality6.6 Tutte's flow conjecturesExercisesNotes 7: Extremal Graph Theory7.1 Subgraphs* 7.2 Minors(*) 7.3 Hadwiger's conjecture*7.4 Szemeredi's regularity lemma7.5 Applying the regularity lemmaExercisesNotes8: Infinite Graphs8.1 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 Notes9: Ramsey Theory for Graphs9.1 Ramsey's original theorems*9.2 Ramsey numbers(*)9.3 Induced Ramsey theorems9.4 Ramsey properties and connectivity(*)ExercisesNotes10: Hamilton Cycles10.1 Simple sufficient conditions*10.2 Hamilton cycles and degree sequences*10.3 Hamilton cycles in the square of a graphExercisesNotes11: Random Graphs11.1 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 momentsExercisesNotes 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 NotesA. 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.