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

Abstract

Resolving a conjecture of Espuny Díaz and Girão [Random Structures Algorithms, 2023], we determine the sharp minimum-degree threshold for Hamiltonicity in graphs perturbed by a uniformly random $K_r$-factor. In fact, we prove the stronger pancyclic statement. More generally, for each fixed connected graph $F$, we study the union of an arbitrary deterministic graph of linear minimum degree and a uniformly random $F$-factor. Let $α^*(F)$ and $α_{\text{pan}}^*(F)$ denote the corresponding Hamiltonicity and pancyclicity thresholds. We introduce two new parameters, $τ_{\text{pc}}(F)$ and $τ_{\text{ind}}(F)$, defined by the expected path-cover number and independence number of random induced subgraphs of $F$, and prove \[ τ_{\text{pc}}(F)\le α^*(F)\le α_{\text{pan}}^*(F)\le τ_{\text{ind}}(F). \] For $F=K_r$, the two parameters coincide and are equal to the unique positive solution $ρ_r$ of $x^r+rx-1=0$. Hence $α^*(K_r)=α_{\text{pan}}^*(K_r)=ρ_r$ for every $r\ge2$.