Resilient pancyclicity of random and pseudorandom graphs

A graph G on n vertices is pancyclic if it contains cycles of length t for all 3 ≤ t ≤ n. In this paper we prove that for any fixed ε > 0, the random graph G(n, p) with p(n) » n^-1/2 (i.e., with p(n)/n^-1/2 tending to infinity) asymptotically almost surely has the following resilience property. If H is a subgraph of G withmaximumdegree at most (1/2-ε)np, then G-H is pancyclic. In fact, we prove a more general result which says that if p » n^-1+1/(l-1) for some integer l ≥ 3, then for any ε > 0, asymptotically almost surely every subgraph of G(n, p) with minimum degree greater than (1/2 + ε)np contains cycles of length t for all l ≤ t ≤ n. These results are tight in two ways. First, the condition on p essentially cannot be relaxed. Second, it is impossible to improve the constant 1/2 in the assumption for the minimum degree. We also prove corresponding results for pseudorandom graphs.