The period since the publication of the first edition of this book has seen the theory of random graphs go from strength to strength. Indeed, its appearance happened to coincide with a watershed in the subject; the emergence in the subsequent few years of singnificant new ideas and tools, perhaps most noteably concentration methods, has had a major impact. It could be argued that the subject is now qualitatively different, insofar as results which would previously have been inaccessible are now regarded as routine. Several long standing issues have been resolved, including the value of the chromatic number of a random graph $G-{n,p}$, the existence of Hamilton cycles in random cubic graphs, and precise bounds on certain Ramsey numbers. It remains the case, though, that most of the material in the first edition of the book is vital for gaining an insight into the theory of random graphs.
Preface
Notation
1 Probability Theoretic Preliminaries
1.1 Notation and Basic Facts
1.2 Some Basic Distributions
1.3 Normal Approximation
1.4 Inequalities
1.5 Convergence in Distribution
2 Models of Random Graphs
2.1 The Basic Models
2.2 Properties of Almost All Graphs
2.3 Large Subsets of Vertices
2.4 Random Regular Graphs
3 The Degree Sequence
3.1 The Distribution of an Element of the Degree Sequence
3.2 Almost Determined Degrees
3.3 The Shape of the Degree Sequence
3.4 Jumps and Repeated Values
3.5 Fast Algorithms for the Graph Isomorphism Problem
4 Small Subgraphs
4.1 Strictly Balanced Graphs
4.2 Arbitrary Subgraphs
4.3 Poisson Approximation
5 The Evolution of Random Graphs-Spare Components
5.1 Trees of Given Sizes As Components
5.2 The Number of Vertices on Tree Components
5.3 The Largest Tree Components
5.4 Components Containing Cycles
6 The Evolution of Random Graphs-the Giant Component
6.1 A Gap in the Sequence of Components
6.2 The Emergence of the Giant Component
6.3 Small Components after Time
6.4 Further Results
6.5 Two Applications
7 Connectivity and Matchings
7.1 The Connectedness of Random Graphs
7.2 The k-Gonnectedness of Random Graphs
7.3 Matchings in Bipartite Graphs
7.4 Matchings in Random Craphs
7.5 Reliable Networks
7.6 Random Regular Graphs
8 Long Paths and Cycles
9 The Automorphism Group
10 The Diameter
11 Cliques,Independent Sets and Colouring
12 Ramsey Theory
13 Explicit Constructions
14 Sequences,Matrices and Permutations
15 Sorting Algorithms
16 Random Graphs of Small Order
References
Index
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