The perturbation threshold of degenerate graphs

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.