Extreme degrees in random subgraphs of regular graphs

Abstract

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 G p (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 G p (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 G p (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 G p (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.