Book contents
- Frontmatter
- Contents
- Preface to the Second Edition
- Preface
- Notation
- 1 Probability Theoretic Preliminaries
- 2 Models of Random Graphs
- 3 The Degree Sequence
- 4 Small Subgraphs
- 5 The Evolution of Random Graphs—Sparse Components
- 6 The Evolution of Random Graphs—the Giant Component
- 7 Connectivity and Matchings
- 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
7 - Connectivity and Matchings
Published online by Cambridge University Press: 29 March 2011
- Frontmatter
- Contents
- Preface to the Second Edition
- Preface
- Notation
- 1 Probability Theoretic Preliminaries
- 2 Models of Random Graphs
- 3 The Degree Sequence
- 4 Small Subgraphs
- 5 The Evolution of Random Graphs—Sparse Components
- 6 The Evolution of Random Graphs—the Giant Component
- 7 Connectivity and Matchings
- 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
Summary
Perhaps the most basic property of a graph is that of being connected. Thus it is not surprising that the study of connectedness of a r.g. has a vast literature. In fact, for fear of upsetting the balance of the book we cannot attempt to give an account of all the results in the area.
Appropriately, the very first random graph paper of Erdős and Rényi (1959) is devoted to the problem of connectedness, and so are two other of the earliest papers on r.gs: Gilbert (1959) and Austin et al. (1959). Erdős and Rényi proved that (n/2) log n is a sharp threshold function for connectedness. Gilbert gave recurrence formulae for the probability of connectedness of Gp (see Exx. 1 and 2). S. A. Stepanov (1969a, 1970a, b) and Kovalenko (1971) extended results of Erdős and Rényi to the model G{n, (pij)}, and Kelmans (1967a) extended the recurrence formulae of Gilbert. Other extensions are due to Ivchenko (1973b, 1975), Ivchenko and Medvedev (1973), Kordecki (1973) and Kovalenko (1975). In §1 we shall present some of these results in the context of the evolution of random graphs.
We know from Chapter 6 that a.e. graph process is such that a giant component appears shortly after time n/2, and the number of vertices not on the giant component decreases exponentially.
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- Random Graphs , pp. 160 - 200Publisher: Cambridge University PressPrint publication year: 2001
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