Random Graphs

* Ample exercises, figures, and bibliographic references

SVANTE JANSON, PhD, is Professor of Mathematics at Uppsala University, Sweden. <p> TOMASZ LUCZAK, PhD, is Professor of Mathematics at Adam Mickiewicz University, Poland, and a visiting professor at Emory University, Atlanta, Georgia. <p> J RUCINSKI, PhD, is Professor of Mathematics at Adam Mickiewicz University and a visiting professor at Emory University.

Preface

v

Preliminaries

1

(24)

Models of random graphs

1

(5)

Notes on notation and more

6

(6)

Monotonicity

12

(2)

Asymptotic equivalence

14

(4)

Thresholds

18

(2)

Sharp thresholds

20

(5)

Exponentially Small Probabilities

25

(28)

Independent summands

26

(4)

Binomial random subsets

30

(4)

Suen's inequality

34

(3)

Martingales

37

(2)

Talagrand's inequality

39

(9)

The upper tail

48

(5)

Small Subgraphs

53

(28)

The containment problem

53

(9)

Leading overlaps and the subgraphs plot

62

(4)

Subgraph count at the threshold

66

(2)

The covring problem

68

(7)

Disjoint copies

75

(2)

Variations on the theme

77

(4)

Matchings

81

(22)

Perfect matchings

82

(7)

G-factors

89

(7)

Two open problems

96

(7)

The Phase Transition

103

(36)

The evolution of the random graph

103

(4)

The emergence of the giant component

107

(5)

The emergence of the giant: A closer look

112

(9)

The structure of the giant component

121

(5)

Near the critical period

126

(2)

Global properties and the summetry rule

128

(6)

Dynamic properties

134

(5)

Asymptotic Distributions

139

(40)

The method of modments

140

(12)

Stein's method: The Poisson case

152

(5)

Stein's method: The normal case

157

(5)

Projections and decompositions

162

(14)

Further methods

176

(3)

The Chromatic number

179

(22)

The stability number

179

(5)

The chromatic number: A greedy approach

184

(3)

The concentration of the chromatic number

187

(3)

The chromatic number of dense random graphs

190

(2)

The chromatic number of sparse random graphs

192

(4)

Vertex partition properties

196

(5)

Extremal and Ramsey Properties

201

(32)

Heuristics and results

202

(7)

Triangles: The first approach

209

(3)

The Szemeredi Regularity lemma

212

(4)

A partition theorem for random graphs

216

(6)

Triangles: An approach with perspective

222

(11)

Random Regular Graphs

233

(38)

The configuration model

235

(1)

Small cycles

236

(3)

Hamilton cycles

239

(8)

Proofs

247

(9)

Contiguity of random regular graphs

256

(8)

A brief course in contiguity

264

(7)

Zero-One Laws

271

(36)

Preliminaries

271

(2)

Ehrenfeucht games and zero-one laws

273

(12)

Filling gaps

285

(7)

Sums of models

292

(9)

Separability and the speed of convergence

301

(6)

References

307

(20)

Index of Notation

327

(4)

Index

331

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