Pancyclicity of graphs perturbed by a random F-factor

Resolving a conjecture of Espuny D\'iaz and Gir\~ao [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 $\alpha^*(F)$ and $\alpha_{\text{pan}}^*(F)$ denote the corresponding Hamiltonicity and pancyclicity thresholds. We introduce two new parameters, $\tau_{\text{pc}}(F)$ and $\tau_{\text{ind}}(F)$, defined by the expected path-cover number and independence number of random induced subgraphs of $F$, and prove \[ \tau_{\text{pc}}(F)\le \alpha^*(F)\le \alpha_{\text{pan}}^*(F)\le \tau_{\text{ind}}(F). \] For $F=K_r$, the two parameters coincide and are equal to the unique positive solution $\rho_r$ of $x^r+rx-1=0$. Hence $\alpha^*(K_r)=\alpha_{\text{pan}}^*(K_r)=\rho_r$ for every $r\ge2$.