The perturbation threshold of degenerate graphs
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arXiv
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| Auteurs principaux: | , , , |
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| Format: | Preprint |
| Publié: |
2026
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| _version_ | 1866917294113816576 |
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| author | Han, Jie Im, Seonghyuk Wang, Bin Zhang, Junxue |
| author_facet | Han, Jie Im, Seonghyuk Wang, Bin Zhang, Junxue |
| contents | We show that for any $d\ge 2$ and $Δ>0$ there exists $η>0$ such that the following holds: Let $G$ be an $n$-vertex graph with at least $Ω(n^2)$ edges and let $H$ be an $n$-vertex $d$-degenerate graph with maximum degree at most $Δ$. Then with high probability, $G \cup G(n, n^{-1/d - η})$ contains a copy of $H$. We also prove that the same conclusion extends to $d$-regular graphs with $d\ge 4$ satisfying a certain edge expansion property, with the threshold improved to $n^{-2/d - η}$. Such a property is satisfied by almost all
$d$-regular graphs and for even $d$, by the $(d/2)$-th power of a Hamilton cycle. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_21867 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | The perturbation threshold of degenerate graphs Han, Jie Im, Seonghyuk Wang, Bin Zhang, Junxue Combinatorics We show that for any $d\ge 2$ and $Δ>0$ there exists $η>0$ such that the following holds: Let $G$ be an $n$-vertex graph with at least $Ω(n^2)$ edges and let $H$ be an $n$-vertex $d$-degenerate graph with maximum degree at most $Δ$. Then with high probability, $G \cup G(n, n^{-1/d - η})$ contains a copy of $H$. We also prove that the same conclusion extends to $d$-regular graphs with $d\ge 4$ satisfying a certain edge expansion property, with the threshold improved to $n^{-2/d - η}$. Such a property is satisfied by almost all $d$-regular graphs and for even $d$, by the $(d/2)$-th power of a Hamilton cycle. |
| title | The perturbation threshold of degenerate graphs |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2602.21867 |