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arXiv · 2602.21867

The perturbation threshold of degenerate graphs

Jie Han·Seonghyuk Im·Bin Wang·Junxue Zhang
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Abstract

We show that for any $d\ge 2$ and $\Delta>0$ there exists $\eta>0$ such that the following holds: Let $G$ be an $n$-vertex graph with at least $\Omega(n^2)$ edges and let $H$ be an $n$-vertex $d$-degenerate graph with maximum degree at most $\Delta$. Then with high probability, $G \cup G(n, n^{-1/d - \eta})$ 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 $n^{-2/d - \eta}$. 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.

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Jie Han, Seonghyuk Im, Bin Wang, Junxue Zhang. 2026-02-25. The perturbation threshold of degenerate graphs. https://arxiv.org/abs/2602.21867

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