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 +ϵn) log n/n)1/2 where ϵn = (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)).
ASJC Scopus subject areas
- Geometry and Topology