The threshold probability for long cycles

For a given graph G of minimum degree at least k, let Gp denote the random spanning subgraph of G obtained by retaining each edge independently with probability p = p(k). We prove that if p ≥ log k+log log k+ωk(1)k, where ωk(1) is any function tending to infinity with k, then Gp asymptotically almost surely contains a cycle of length at least k + 1. When we take G to be the complete graph on k + 1 vertices, our theorem coincides with the classic result on the threshold probability for the existence of a Hamilton cycle in the binomial random graph.