图论导引 第2版 英文版【2025|PDF下载-Epub版本|mobi电子书|kindle百度云盘下载】

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Chapter 1 Fundamental Concepts1

1.1 What Is a Graph？1

The Definition1

Graphs as Models3

Matrices and Isomorphism6

Decomposition and Special Graphs11

Exercises14

1.2 Paths,Cycles,and Trails19

Connection in Graphs20

Bipartite Graphs24

Eulerian Circuits26

Exercises31

1.3 Vertex Degrees and Counting34

Counting and Bijections35

Extremal Problems38

Graphic Sequences44

Exercises47

1.4 Directed Graphs53

Definitions and Examples53

Vertex Degrees58

Eulerian Digraphs60

Orientations and Tournaments61

Exercises63

Chapter 2 Trees and Distance67

2.1 Basic Properties67

Properties of Trees68

Distance in Trees and Graphs70

Disjoint Spanning Trees(optional)73

Exercises75

2.2 Spanning Trees and Enumeration81

Enumeration of Trees81

Spanning Trees in Graphs83

Decomposition and Graceful Labelings87

Branchings and Eulerian Digraphs(optional)89

Exercises92

2.3 Optimization and Trees95

Minimum Spanning Tree95

Shortest Paths97

Trees in Computer Science(optional)100

Exercises103

Chapter 3 Matchings and Factors107

3.1 Matchings and Covers107

Maximum Matchings108

Hall's Matching Condition110

Min-Max Theorems112

Independent Sets and Covers113

Dominating Sets(optional)116

Exercises118

3.2 Algorithms and Applications123

Maximum Bipartite Matching123

Weighted Bipartite Matching125

Stable Matchings(optional)130

Faster Bipartite Matching(optional)132

Exercises134

3.3 Matchings in General Graphs136

Tutte's 1-factor Theorem136

f-factors of Graphs(optional)140

Edmonds'Blossom Algorithm(optional)142

Exercises145

Chapter 4 Connectivity and Paths149

4.1 Cuts and Connectivity149

Connectivity149

Edge-connectivity152

Blocks155

Exercises158

4.2 k-connected Graphs161

2-connected Graphs161

Connectivity of Digraphs164

k-connected and k-edge-connected Graphs166

Applications of Menger's Theorem170

Exercises172

4.3 Network Flow Problems176

Maximum Network Flow176

Integral Flows181

Supplies and Demands(optional)184

Exercises188

Chapter 5 Coloring of Graphs191

5.1 Vertex Colorings and Upper Bounds191

Definitions and Examples191

Upper Bounds194

Brooks'Theorem197

Exercises199

5.2 Structure of k-chromatic Graphs204

Graphs with Large Chromatic Number205

Extremal Problems and Turán's Theorem207

Color-Critical Graphs210

Forced Subdivisions212

Exercises214

5.3 Enumerative Aspects219

Counting Proper Colorings219

Chordal Graphs224

A Hint of Perfect Graphs226

Counting Acyclic Orientations(optional)228

Exercises229

Chapter 6 Planar Graphs233

6.1 Embeddings and Euler's Formula233

Drawings in the Plane233

Dual Graphs236

Euler's Formula241

Exercises243

6.2 Characterization of Planar Graphs246

Preparation for Kuratowski's Theorem247

Convex Embeddings248

Planarity Testing(optional)252

Exercises255

6.3 Parameters of Planarity257

Coloring of Planar Graphs257

Crossing Number261

Surfaces of Higher Genus(optional)266

Exercises269

Chapter 7 Edges and Cycles273

7.1 Line Graphs and Edge-coloring273

Edge-colorings274

Characterization of Line Graphs(optional)279

Exercises282

7.2 Hamiltonian Cycles286

Necessary Conditions287

Sufficient Conditions288

Cycles in Directed Graphs(optional)293

Exercises294

7.3 Planarity,Coloring,and Cycles299

Tait's Theorem300

Grinberg's Theorem302

Snarks(optional)304

Flows and Cycle Covers(optional)307

Exercises314

Chapter 8 Additional Topics(optional)319

8.1 Perfect Graphs319

The Perfect Graph Theorem320

Chordal Graphs Revisited323

Other Classes of Perfect Graphs328

Imperfect Graphs334

The Strong Perfect Graph Conjecture340

Exercises344

8.2 Matroids349

Hereditary Systems and Examples349

Properties of Matroids354

The Span Function358

The Dual of a Matroid360

Matroid Minors and Planar Graphs363

Matroid Intersection366

Matroid Union369

Exercises372

8.3 Ramsey Theory378

The Pigeonhole Principle Revisited378

Ramsey's Theorem380

Ramsey Numbers383

Graph Ramsey Theory386

Sperner's Lemma and Bandwidth388

Exercises392

8.4 More Extremal Problems396

Encodings of Graphs397

Branchings and Gossip404

List Coloring and Choosability408

Partitions Using Paths and Cycles413

Circumference416

Exercises422

8.5 Random Graphs425

Existence and Expectation426

Properties of Almost All Graphs430

Threshold Functions432

Evolution and Graph Parameters436

Connectivity,Cliques,and Coloring439

Martingales442

Exercises448

8.6 Eigenvalues of Graphs452

The Characteristic Polynomial453

Linear Algebra of Real Symmetric Matrices456

Eigenvalues and Graph Parameters458

Eigenvalues of Regular Graphs460

Eigenvalues and Expanders463

Strongly Regular Graphs464

Exercises467

Appendix A Mathematical Background471

Sets471

Quantifiers and Proofs475

Induction and Recurrence479

Functions483

Counting and Binomial Coefficients485

Relations489

The Pigeonhole Principle491

Appendix B Optimization and Complexity493

Intractability493

Heuristics and Bounds496

NP-Completeness Proofs499

Exercises505

Appendix C Hints for Selected Exercises507

General Discussion507

Supplemental Specific Hints508

Appendix D Glossary of Terms515

Appendix E Supplemental Reading533

Appendix F References567

Author Index569

Subject Index575