Cycle factors in randomly perturbed graphs

We study the problem of finding pairwise vertex-disjoint copies of the $\mathrm{\ell }$-vertex cycle $C_{\mathrm{\ell }}$ 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 $\mathrm{\ell }\ge 3$ we prove that asymptotically almost surely $G \cup G(n,p)$contains$\min {\delta(G), \lfloor n/\ell \rfloor }$ pairwise vertex-disjoint cycles $C_{\mathrm{\ell }}$, provided $p\ge C\log n\mathrm{/}n$ for $C$ sufficiently large. Moreover, when $\delta (G)\ge \alpha n$ with $0<\alpha \le 1/\ell$and$G$isnot\mathrm{‘}clos{e}^{\mathrm{\prime }}tothecompletebipartitegraph$K_{\alpha n,(1-\alpha) n}$,then$p \ge C/n$ suffices to get the same conclusion. This provides a stability version of our result. In particular, we conclude that $p \ge C/n$sufficeswhen$\alpha>1/\ell$forfinding$\lfloor n/\ell \rfloor$cycles$C_\ell$. Our results are asymptotically optimal. They can be seen as an interpolation between the Johansson–Kahn–Vu Theorem for $C_{\mathrm{\ell }}$-factors and the resolution of the El-Zahar Conjecture for $C_{\mathrm{\ell }}$-factors by Abbasi.