Local connectivity of a random graph

A graph is locally connected if for each vertex ν of degree ≧2 , the subgraph induced by the vertices adjacent to ν 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 + ϵ n ) log n/n) 1/2 where ϵ n = (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)).