On Hamilton cycles in Erdos-Renyi subgraphsof large graphs

摘要

Given a graph Gamma(n)=(V,E) on n vertices and m edges, we define the Erdos-Renyi graph process with host Gamma(n) as follows. A permutation e(1), horizontal ellipsis ,e(m) of E is chosen uniformly at random, and for t <= m we let Gamma(n,t)=(V,{e(1), horizontal ellipsis ,e(t)}). Suppose the minimum degree of Gamma(n) is delta(Gamma(n)) >= (1/2+epsilon)n for some constant epsilon>0. Then with high probability (An event & x2130;n holds with high probability (whp) if Pr & x2130;n -> 1 as n ->infinity.), Gamma(n,t) becomes Hamiltonian at the same moment that its minimum degree reaches 2. Given 0 <= p <= 1 let Gamma(n,p) be the Erdos-Renyi subgraph of Gamma(n), obtained by retaining each edge independently with probability p. When delta(Gamma(n)) >= (1/2+epsilon)n, we provide a threshold for Hamiltonicity in Gamma(n,p).