Long Paths and Cycles in Random Subgraphs of H-free Graphs

Let H be a given (possibly empty) family of connected graphs, each containing a cycle, and let G be a arbitrary finite H-free graph with minimum degree at least k. For p ∈ [0, 1], we form the p-random subgraph Gp of G by independently keeping each edge of G with probability p. Extending a classical result of Ajtai, Komlós, and Szemerédi, we prove that for every positive ε, there exists a positive δ (depending only on ε) such that the following holds: If p > 1+ε k , then with probability tending to 1 as k → ∞, the random graph Gp contains a cycle of length at least δ ·nH(k), where nH(k) > k is the minimum number of vertices in an H-free graph of average degree at least k. Thus in particular Gp as above typically contains a cycle of length at least linear in k.