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Extreme degrees in random subgraphs of regular graphs

Published online by Cambridge University Press:  24 October 2008

Zbigniew Palka
Zbigniew Palka
Affiliation:
Institute of Mathematics, A. Mickiewicz University, Poznań, Poland
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Let G(d) be a given simple d-regular graph on n labelled vertices, where dn is even. Such a graph will be called an initial graph. Denote by Gp(d) a random subgraph of G(d) obtained by removing edges, each with the same probability q — 1 —p, independently of all other edges (i.e. each edge remains in Gp(d) with probability p). In a recent paper [10] the asymptotic distributions of the number of vertices of a given degree in a random graph Gp(d) were given. The aim of this sequel is to present a wide variety of results devoted to probability distributions of the maximum and minimum degree of Gp(d) with respect to different values of the edge probability p and degree of regularity d. It should be noted here that very detailed results on a similar subject in the case when the initial graph is a complete graph (i.e. when d = n – 1) have already been obtained by Bollobás in the series of papers [2]–[4] (some additional information to the paper [4] was given in [9]). Also, in proving our results we will make use of some ideas given by Bollobás in these papers.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 97 , Issue 1 , January 1985 , pp. 69 - 78
Copyright
Copyright © Cambridge Philosophical Society 1985

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References

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