グラフ理論(第4版)<br>Graph Theory (Graduate Texts in Mathematics) 〈Vol. 173〉 (4TH)

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グラフ理論(第4版)
Graph Theory (Graduate Texts in Mathematics) 〈Vol. 173〉 (4TH)

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  • 製本 Paperback:紙装版/ペーパーバック版/ページ数 410 p.
  • 言語 ENG
  • 商品コード 9783642142789

基本説明

現代グラフ理論の定番テキスト、4年振りの改訂版。
This standard textbook of modern graph theory, now in its fourth 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.

Full Description


Almosttwodecadeshavepassedsincetheappearanceofthosegrapht- ory texts that still set the agenda for most introductory courses taught today. The canon created by those books has helped to identify some main?eldsofstudyandresearch,andwilldoubtlesscontinuetoin?uence the development of the discipline for some time to come. Yet much has happened in those 20 years, in graph theory no less thanelsewhere: deepnewtheoremshavebeenfound,seeminglydisparate methods and results have become interrelated, entire new branches have arisen. To name just a few such developments, one may think of how the new notion of list colouring has bridged the gulf between inva- ants such as average degree and chromatic number, how probabilistic methods andtheregularity lemmahave pervadedextremalgraphtheory and Ramsey theory, or how the entirely new ?eld of graph minors and tree-decompositions has brought standard methods of surface topology to bear on long-standing algorithmic graph problems. Clearly, then, the time has come for a reappraisal: what are, today, the essential areas, methods and results that should form the centre of an introductory graph theory course aiming to equip its audience for the most likely developments ahead? I have tried in this book to o?er material for such a course. In view of the increasing complexity and maturity of the subject, I have broken with the tradition of attempting to cover both theory and app- cations: this book o?ers an introduction to the theory of graphs as part of (pure) mathematics; it contains neither explicit algorithms nor `real world' applications.

Table of Contents

Preface                                            vii
1 The Basics 1 (34)
1.1 Graphs* 2 (3)
1.2 The degree of a vertex* 5 (1)
1.3 Paths and cycles* 6 (4)
1.4 Connectivity* 10 (3)
1.5 Trees and forests* 13 (4)
1.6 Bipartite graphs* 17 (2)
1.7 Contraction and minors* 19 (3)
1.8 Euler tours* 22 (1)
1.9 Some linear algebra 23 (5)
1.10 Other notions of graphs 28 (2)
Exercises 30 (3)
Notes 33 (2)
2 Matching Covering and Packing 35 (24)
2.1 Matching in bipartite graphs* 36 (5)
2.2 Matching in general graphs(*) 41 (4)
2.3 Packing and covering 45 (3)
2.4 Tree-packing and arboricity 48 (4)
2.5 Path covers 52 (2)
Exercises 54 (2)
Notes 56 (3)
3 Connectivity 59 (28)
3.1 2.Connected graphs and subgraphs* 59
3.2 The structure of 3-connected graphs(*) 02 (64)
3.3 Menger's theorem* 66 (6)
3.1 Mader's theorem 72 (2)
3.5 Linking pails of vertices(*) 74 (8)
Exercises 82 (3)
Notes 85 (2)
4 Planar Graphs 87 (30)
4.1 Topological prerequisites* 88 (2)
4.2 Platte graphs* 90 (6)
4.3 Drawings 96 (94)
4.4 Planar graphs: Kuratowski's theorem* 190
4.5 Algebraic planarity criteria 105(2)
4.6 Plane duality 107(4)
Exercises 111(3)
Notes 114(3)
5 Colouring 117(28)
5.1 Colouring maps and planar graphs* 118(2)
5.2 Colouring vertices* 120(5)
5.3 Colouring edges* 125(2)
5.1 List colouring 127(5)
5.5 Perfect graphs 132(7)
Exercises 139(4)
Notes 143(2)
6 Flows 145(24)
6.1 Circulatins(*) 146(1)
6.2 Flows in networks* 147(3)
6.3 Group-valued flows 150(5)
6.4 k-Flows for small k 155(3)
6.5 Flow-colouring duality 158(3)
6.6 Tutte's flow conjectures 161(4)
Exercise 165(2)
Notes 167(2)
7 Extremal Graph Theory 169(34)
7.1 Subgraphs* 170(5)
7.2 Minors(*) 175(3)
7.3 Hadwiger's conjecture* 178(4)
7.4 Szemer馘i's regularity lemma 182(7)
7.5 Applying the regularity lemma 189(6)
Exercises 195(3)
Notes 198(5)
8 Infinite Graphs 203(66)
8.1 Basic notions, facts and techniques* 204(9)
8.2 Paths, trees, and ends(*) 213(9)
8.3 Homogeneous and universal graphs* 222(3)
8.4 Connectivity and matching 225(10)
8.5 Graphs with ends: the topological 235(13)
viewpoint
8.6 Recursive structures 248(3)
Exercises 251(10)
Notes 261(8)
9 Ramsey Theory for Graphs 269(24)
9.1 Ramsey's original theorems* 270(3)
9.2 Ramsey numbers(*) 273(3)
9.3 Induced Ramsey theorems 276(10)
9.4 Ramsey properties and connectivity(*) 286(3)
Exercises 289(1)
Notes 290(3)
10 Hamilton Cycles 293(16)
10.1 Sufficient conditions* 293(4)
10.2 Hamilton cycles and degree sequences* 297(3)
10.3 Hamilton cycles in the square of a graph 300(5)
Exercises 305(1)
Notes 306(3)
11 Random Graphs 309(24)
11.1 The notion of a random graph* 310(5)
11.2 The probabilistic method* 315
11.3 Properties of almost all graphs* 118(204)
11.4 Threshold functions and second moments 322(7)
Exercises 329(1)
Notes 330(3)
12 Minors, Trees arid WQO 333(44)
12.1 Well-quasi-ordering* 334(1)
12.2 The graph minor theorem for trees* 335(2)
12.3 Tree-decompositions 337(8)
12.4 nee-width and, forbidden minors 345(14)
12.5 The graph minor theorem(*) 359(9)
Exercises 368(5)
Notes 373(4)
A Infinite sets 377(6)
B Surfaces 383(8)
Hints for all the exercises 391(28)
Index 419(16)
Symbol index 435