Pancyclic Hamilton cycles in random graphs

Abstract

Let G ( n,p) denote the probability space of the set G of graphs G = ( V n , E) with vertex set V n = {1,2,…, n} and edges E chosen independently with probability p from E={{u,v}:u,v∈V n,u≠v} . A graph G∈ G ( n,p is defined to be pancyclic if, for all s, 3⩽ s⩽ n there is a cycle of size s on the edges of G. We show that the threshold probability p = ( log n + log log n + c n )/ n for the property that G contains a Hamilton cycle is also the threshold probability for the existence of a 2-pancyclic Hamilton cycle, which is defined as follows. Given a Hamilton cycle H, we will say that H is k-pancyclic if for each s (3⩽ s⩽ n−1) we can find a cycle C of length s using only the edges of H and at most k other edges.