Pancyclicity of graphs perturbed by a random $F$-factor

Abstract

We determine the sharp minimum-degree threshold for Hamiltonicity in graphs perturbed by a uniformly random $K_r$-factor, resolving a conjecture of Espuny Díaz and Girão [Random Structures Algorithms, 2023]. In fact, we prove the stronger pancyclic statement. Let $α^*(K_r)$ and $α_{\text{pan}}^*(K_r)$ denote the Hamiltonicity and pancyclicity thresholds, respectively. We show that $α^*(K_r)=α_{\text{pan}}^*(K_r)=ρ_r$, where $ρ_r$ is the unique positive solution of $x^r+rx-1=0$. The proof is obtained from a general framework for perturbations by a uniformly random $F$-factor, where $F$ is an arbitrary fixed connected graph.