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 $\hat{I}{\pm }^{\ast }(K_r)$ and $\hat{I}{\pm }_{\text{pan}}^{\ast }(K_r)$ denote the Hamiltonicity and pancyclicity thresholds, respectively. We show that $\hat{I}{\pm }^{\ast }(K_r)=\hat{I}{\pm }_{\text{pan}}^{\ast }(K_r)=\ddot{I}{}_r$, where $\ddot{I}{}_r$ 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.

12 pages