The diameter of random regular graphs

We give asymptotic upper and lower bounds for the diameter of almost every r-regular graph on n vertices (n~). Though random graphs of various types have been investigated extensively over the last twenty years, random regular graphs have hardly been studied. The reason for this is that until recently there was no formula for the asymptotic number of labelled r-regular graphs of order n. Such a formula was given by Bender and Can-field [1]. Even more recently one of the present authors [3] gave a simpler proof of the same formula. More importantly, [3] contains a model tbr the set of regular graphs which can be used to study labelled random regular graphs. Our aim is to investigate the diameter: we shall show that for a fixed r most r-regular graphs of order n have about the same diameter. Our results have some bearing on certain extremal problems concerning graphs of small diameter and small maximum degree (see [2, Ch. IV]). For results about the diameter of the customary random graphs see [4], [5] and [6]. Let us start with the model mentioned above. Let r~3 be fixed and denote by G(n, r-reg) the probability space of all r-regular graphs with a fixed set of n labelled vertices. Here we assume that rn is even and any two graphs have the same probability. We shall say that almost every (a.e.) r-regular graph has a certain property if the probability that a member of G(n, r-reg) has this property tends to 1 as n~ ~. n W = U W~ is a partition of W into pairs. We call these pairs the edges of the configu-1 ration and we denote by ~ the set of all configurations with vertex set W. Once again we view ~2 as a probability space in which all points are equiprobable. Given a configuration FE(2 we may try to construct an r-regular graph q~(F) with vertex set {W~, W2 ..... W,,} as follows. Join two vertices W,. and Wj by an edge iff the configuration Fcontains an edge having one vertex in W~ and the other in Wj.. Clearly q~(F) is an r-regular graph if F has no edge joining two vertices of the same class W~, nor has it two edges joining vertices in the same two classes W~ and W i. All we need AMS subject classification (1980): 05 C 99, 60 C 05.