Local connectivity of a random graph

A graph is locally connected if for each vertex v of degree --2, the subgraph induced by the vertices adjacent to v is connected . In this paper we establish a sharp threshold function for local connectivity . Specifically, if the probability of an edge of a labeled graph of order n is p = ((3/2 + s„) log n/n)'/ 2 where s„ _ ( log log n + log(3/8) + 2x)/(2 log n), then the limiting probability that a random graph is locally connected is exp(exp(-x)) . INTRODUCTION A graph is locally connected if for each vertex v of degree --2, the subgraph induced by the vertices adjacent to v is connected. Our sample space, denoted ,f2,, consists of all 2(2 ) labeled graphs G, if order n . Let p = p(n) be a number between 0 and 1 . If G has size q, i .e ., q is the number of edges, then the probability of G is defined by P(G) = pq(1 P)(z)-q, P . Erdős HUNGARIAN ACADEMY OF SCIENCES, BUDAPEST, HUNGARY E . M . Palmer MICHIGAN STATE UNIVERSITY, EAST LANSING, MI 48824 R . W . Robinson SOUTHERN ILLINOIS UNIVERSITY, CARBONDALE, IL 62901 (1) and we refer to p as the probability of an edge . Our aim is to study the limiting probability, lim,. P(C), where C is the subset of fl, which consists of the locally connected graphs . If p is fixed, then the methods of [BIH79] show immediately that the limit above Journal of Graph Theory, Vol . 7 (1983) 411-417 © 1983 by John Wiley & Sons, Inc . CCC 0364-9024/83/040411-07$01 .70 412 JOURNAL OF GRAPH THEORY is 1, i .e ., "almost all graphs are locally connected ." Therefore we will consider only p(n) such that limn-p(n) = 0 . Note that local connectivity is not a monotone property . That is, if G is locally connected and H is obtained by adding an edge to G, then H is not necessarily locally connected whether G is connected or not . Nevertheless, we find two threshold functions for local connectivity . The first and less interesting occurs when the probability of an edge is so small that almost every graph consists of isolated edges and vertices . But the second threshold is non-trivial and appears when the probability of an edge is sufficiently high to cause every edge to belong to a triangle . For the most part, our notation and terminology follow that of the book [Bo79] where one can also find an introduction to the methods used here . We also assume some familiarity with the fundamental results of the senior co-author in [ErR59] and [ErR60] .