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| Format: | Preprint |
| Veröffentlicht: |
2026
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| Online-Zugang: | https://arxiv.org/abs/2605.29553 |
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| _version_ | 1866917549119111168 |
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| author | Ma, Guorui Yan, Zhifei |
| author_facet | Ma, Guorui Yan, Zhifei |
| contents | We determine the sharp threshold for Hamilton cycles in randomly perturbed sparse graphs. For any $α=α(n)=o(1)$, let $G_α$ be an $n$-vertex graph with minimum degree $δ(G_α)\geαn$. We prove that if $$p\ge(1+\varepsilon)\frac{\log(1/α)}{n},$$ then the union $G_α\cup G(n,p)$ is Hamiltonian asymptotically almost surely. This significantly strengthens a recent result of Hahn-Klimroth, Maesaka, Mogge, Mohr, and Parczyk by improving the leading constant from 6 to the optimal value of 1. Crucially, we show that this bound on $p$ is best possible when $αn\rightarrow\infty$, thereby establishing the exact probability threshold for Hamiltonicity in this sparse regime. Our proof relies on a robust random expansion lemma, Pósa's booster lemma, and a sprinkling argument. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_29553 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Sharp threshold for Hamilton cycles in randomly perturbed sparse graphs Ma, Guorui Yan, Zhifei Combinatorics We determine the sharp threshold for Hamilton cycles in randomly perturbed sparse graphs. For any $α=α(n)=o(1)$, let $G_α$ be an $n$-vertex graph with minimum degree $δ(G_α)\geαn$. We prove that if $$p\ge(1+\varepsilon)\frac{\log(1/α)}{n},$$ then the union $G_α\cup G(n,p)$ is Hamiltonian asymptotically almost surely. This significantly strengthens a recent result of Hahn-Klimroth, Maesaka, Mogge, Mohr, and Parczyk by improving the leading constant from 6 to the optimal value of 1. Crucially, we show that this bound on $p$ is best possible when $αn\rightarrow\infty$, thereby establishing the exact probability threshold for Hamiltonicity in this sparse regime. Our proof relies on a robust random expansion lemma, Pósa's booster lemma, and a sprinkling argument. |
| title | Sharp threshold for Hamilton cycles in randomly perturbed sparse graphs |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2605.29553 |