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arXiv · 2606.02160

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

Dingjia Mao·Feihong Yuan·Wenling Zhou
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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 D\'iaz and Gir\~ao [Random Structures Algorithms, 2023]. In fact, we prove the stronger pancyclic statement. Let $\alpha^*(K_r)$ and $\alpha_{\text{pan}}^*(K_r)$ denote the Hamiltonicity and pancyclicity thresholds, respectively. We show that $\alpha^*(K_r)=\alpha_{\text{pan}}^*(K_r)=\rho_r$, where $\rho_r$ 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.

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Dingjia Mao, Feihong Yuan, Wenling Zhou. 2026-06-01. Pancyclicity of graphs perturbed by a random $F$-factor. https://arxiv.org/abs/2606.02160

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