Cycle lengths in randomly perturbed graphs

Abstract

AbstractLet be an ‐vertex graph, where for some . A result of Bohman, Frieze and Martin from 2003 asserts that if , then perturbing via the addition of random edges, a.a.s. yields a Hamiltonian graph. We prove several improvements and extensions of the aforementioned result. In particular, keeping the bound on as above and allowing for , we determine the correct order of magnitude of the number of random edges whose addition to a.a.s. yields a pancyclic graph. Moreover, we prove similar results for sparser graphs, and assuming the correctness of Chvátal's toughness conjecture, we handle graphs having larger independent sets. Finally, under milder conditions, we determine the correct order of magnitude of the number of random edges whose addition to a.a.s. yields a graph containing an almost spanning cycle.