## Modern Graph TheoryThe time has now come when graph theory should be part of the education of every serious student of mathematics and computer science, both for its own sake and to enhance the appreciation of mathematics as a whole. This book is an in-depth account of graph theory, written with such a student in mind; it reflects the current state of the subject and emphasizes connections with other branches of pure mathematics. The volume grew out of the author's earlier book, Graph Theory -- An Introductory Course, but its length is well over twice that of its predecessor, allowing it to reveal many exciting new developments in the subject. Recognizing that graph theory is one of several courses competing for the attention of a student, the book contains extensive descriptive passages designed to convey the flavor of the subject and to arouse interest. In addition to a modern treatment of the classical areas of graph theory such as coloring, matching, extremal theory, and algebraic graph theory, the book presents a detailed account of newer topics, including Szemer\'edi's Regularity Lemma and its use, Shelah's extension of the Hales-Jewett Theorem, the precise nature of the phase transition in a random graph process, the connection between electrical networks and random walks on graphs, and the Tutte polynomial and its cousins in knot theory. In no other branch of mathematics is it as vital to tackle and solve challenging exercises in order to master the subject. To this end, the book contains an unusually large number of well thought-out exercises: over 600 in total. Although some are straightforward, most of them are substantial, and others will stretch even the most able reader. |

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a really bad book for study, since it does not contain any solutions to its exercises.

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The online version of this text is badly distorted as to make it unreadable.

### Contents

Fundamentals | xv |

I2 Paths Cycles and Trees | xxii |

I3 Hamilton Cycles and Euler Circuits | 4 |

I4 Planar Graphs | 10 |

I5 An Application of Euler Trails to Algebra | 15 |

I6 Exercises | 18 |

Electrical Networks | 29 |

II2 Squaring the Square | 36 |

VI3 Ramsey Theory For Graphs | 182 |

VI4 Ramsey Theory for Integers | 187 |

VI5 Subsequences | 195 |

VI6 Exercises | 198 |

VI7 Notes | 203 |

Random Graphs | 205 |

VII1 The Basic ModelsThe Use of the Expectation | 206 |

VII2 Simple Properties of Almost All Graphs | 215 |

II3 Vector Spaces and Matrices Associated with Graphs | 41 |

II4 Exercises | 48 |

II5 Notes | 56 |

Hows Connectivity and Matching | 57 |

III1 Flows in Directed Graphs | 58 |

III2 Connectivity and Mengers Theorem | 63 |

III3 Matching | 66 |

III4 Tuttes 1Factor Theorem | 72 |

III5 Stable Matchings | 75 |

III6 Exercises | 81 |

III7 Notes | 91 |

Extremal Problems | 93 |

IV1 Paths and Cycles | 94 |

IV2 Complete Subgraphs | 98 |

IV3 Hamilton Paths and Cycles | 105 |

IV4 The Structure of Graphs | 110 |

IV5 Szemeredis Regularity Lemma | 114 |

IV6 Simple Applications of Szemeredis Lemma | 120 |

IV7 Exercises | 125 |

IV8 Notes | 132 |

Colouring | 135 |

V1 Vertex Colouring | 136 |

V2 Edge Colouring | 142 |

V3 Graphs on Surfaces | 144 |

V4 List Colouring | 151 |

V5 Perfect Graphs | 155 |

V6 Exercises | 160 |

V7 Notes | 167 |

Ramsey Theory | 171 |

VI1 The Fundamental Ramsey Theorems | 172 |

VI2 Canonical Ramsey Theorems | 179 |

VII3 Almost Determined Variables The Use of the Variance | 218 |

VII4 Hamilton CyclesThe Use of Graph Theoretic Tools | 226 |

VII5 The Phase Transition | 230 |

VII6 Exercises | 236 |

VII7 Notes | 241 |

Graphs Groups and Matrices | 243 |

VIII1 Cayley and Schreier Diagrams | 244 |

VIII2 The Adjacency Matrix and the Laplacian | 252 |

VIII3 Strongly Regular Graphs | 260 |

VIII4 Enumeration and Polyas Theorem | 266 |

VIII5 Exercises | 273 |

Random Walks on Graphs | 285 |

IX1 Electrical Networks Revisited | 286 |

IX2 Electrical Networks and Random Walks | 291 |

IX3 Hitting Times and Commute Times | 299 |

IX4 Conductance and Rapid Mixing | 309 |

IX5 Exercises | 317 |

IX6 Notes | 323 |

The Tutte Polynomial | 325 |

X1 Basic Properties of the Tutte Polynomial | 326 |

X2 The Universal Form of the Tutte Polynomial | 330 |

X3 The Tutte Polynomial in Statistical Mechanics | 332 |

X4 Special Values of the Tutte Polynomial | 335 |

X5 A Spanning Tree Expansion of the Tutte Polynomial | 340 |

X6 Polynomials of Knots and Links | 348 |

X7 Exercises | 361 |

X8 Notes | 367 |

Symbol Index | 369 |

Name Index | 373 |

377 | |

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### Common terms and phrases

adjacency matrix Algebraic algorithm assertion bipartite graph Cayley diagram chromatic number chromatic polynomial Co(G colour combinatorial complete graph components connected graph Corollary Deduce define denote edge xy edges of G eigenvalues electrical network equivalent Erdos ex(n Exercise extremal graph finite flow function G contains Gn,p graph G graph of order graph theory Hamilton cycle Hamiltonian Hence Hint implies independent edges induced subgraph induction inequality integer isomorphic knot least Lemma Let G link diagram loop Math max-flow min-cut theorem maximal number minimal degree monochromatic multigraph natural numbers neighbours number of edges oriented partition perfect graph planar plane graph precisely problem proof of Theorem prove random graph random walks rectangle regular graph resistance result simple spanning forest spanning tree stable matching subgraph of G subsets Suppose Theorem 14 Tr(n trefoil knot triangle Tutte polynomial vector vertex vertex set

### Popular passages

Page vi - As long as a branch of science offers an abundance of problems, so long is it alive; a lack of problems foreshadows extinction or the cessation of independent development. Just as every human undertaking pursues certain objects, so also mathematical research requires its problems. It is by the solution of problems that the investigator tests the temper of his steel; he finds new methods and new outlooks, and gains a wider and freer horizon.