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 > 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 with maximum degree at most (1/2− )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 > 0, asymptotically almost surely every subgraph of G(n, p) with minimum degree greater than (1/2 + )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.