Cycle factors in randomly perturbed graphs

We study the problem of finding pairwise vertex-disjoint copies of the ω>-vertex cycle Cω>in the randomly perturbed graph model, which is the union of a deterministic n-vertex graph G and the binomial random graph G(n, p). For ω>≥ 3 we prove that asymptotically almost surely G U G(n, p) contains min{δ(G), min{δ(G), [n/l]} pairwise vertex-disjoint cycles Cω>, provided p ≥ C log n/n for C sufficiently large. Moreover, when δ(G) ≥ αn with 0 ≤ α/l and G and is not 'close' to the complete bipartite graph Kαn(1 - α)n, then p ≥ C/n suffices to get the same conclusion. This provides a stability version of our result. In particular, we conclude that p ≥ C/n suffices when α > n/l for finding [n/l] cycles Cω>. Our results are asymptotically optimal. They can be seen as an interpolation between the Johansson-Kahn-Vu Theorem for Cω>-factors and the resolution of the El-Zahar Conjecture for Cω>-factors by Abbasi.