LONG PATHS AND CYCLES IN RANDOM SUBGRAPHS OF H-FREE GRAPHS

Abstract. Let H be a given finite (possibly empty) family of connected graphs, each containing a cycle, and let G be an arbitrary finite H-free graph with minimum degree at least k. For p ∈ [0, 1], we form a 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, then k with probability tending to 1 as k → ∞, the random graph Gp contains a cycle of length at least nH(δk), 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. 1