Cycle factors in randomly perturbed graphs

TL;DR

This paper investigates the existence of multiple disjoint cycles in randomly perturbed graphs, establishing thresholds for the probability of adding random edges to guarantee such structures.

Contribution

It provides new thresholds for cycle packings in perturbed graphs, extending known results and offering stability versions for various degree conditions.

Findings

Asymptotically almost surely, the union contains the minimum of degree and floor division by cycle length.

Thresholds for random edge probability are established, e.g., p ≥ C log n / n or p ≥ C / n.

Results interpolate between known theorems on cycle factors and conjectures, showing optimality.

Abstract

We study the problem of finding pairwise vertex-disjoint copies of the $\ell$-vertex cycle $C_\ell$ 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 $\ell \ge 3$ we prove that asymptotically almost surely $G \cup G(n,p)$ contains $\min \{\delta(G), \lfloor n/\ell \rfloor \}$ pairwise vertex-disjoint cycles $C_\ell$, provided $p \ge C \log n/n$ for $C$ sufficiently large. Moreover, when $\delta(G) \ge\alpha n$ with $0<\alpha \le 1/\ell$ and $G$ is not `close' to the complete bipartite graph $K_{\alpha n,(1-\alpha) n}$, then $p \ge C/n$ suffices to get the same conclusion. This provides a stability version of our result. In particular, we conclude that $p \ge C/n$ suffices when $\alpha>1/\ell$ for finding $\lfloor n/\ell \rfloor$ cycles $C_\ell$. Our results are asymptotically optimal. They can be seen as an interpolation between the Johansson--Kahn--Vu Theorem for $C_\ell$-factors and the resolution of the El-Zahar Conjecture for $C_\ell$-factors by Abbasi.