A graph G is said to be pancyclic if G contains cycles of all lengths from 3 to |V (G)|. We show that if G is 4-connected, claw-free, and P10-free, then G is either pancyclic or it is the line graph of the Petersen graph. This implies that every 4-connected, claw-free, P9-free graph is pancyclic, which is best possible and extends a result of Gould et al. Pancyclicity in 3-connected graphs: Pai...

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 m...