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| Main Authors: | , , |
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| Format: | Preprint |
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2026
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| Online Access: | https://arxiv.org/abs/2606.02160 |
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| _version_ | 1866917555279495168 |
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| author | Mao, Dingjia Yuan, Feihong Zhou, Wenling |
| author_facet | Mao, Dingjia Yuan, Feihong Zhou, Wenling |
| contents | 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 $r\ge2$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2606_02160 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Pancyclicity of graphs perturbed by a random $F$-factor Mao, Dingjia Yuan, Feihong Zhou, Wenling Combinatorics 05C38 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 $r\ge2$. |
| title | Pancyclicity of graphs perturbed by a random $F$-factor |
| topic | Combinatorics 05C38 |
| url | https://arxiv.org/abs/2606.02160 |