The perturbation threshold of degenerate graphs

We show that for any $d\ge 2$ and $\mathrm{\Delta }>0$ there exists $\eta >0$ such that the following holds: Let $G$ be an $n$-vertex graph with at least $\mathrm{\Omega }(n^2)$ edges and let $H$ be an $n$-vertex $d$-degenerate graph with maximum degree at most $\mathrm{\Delta }$. Then with high probability, $G\cup G(n,n^{-1\mathrm{/}d-\eta })$ 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 $n^{-2\mathrm{/}d-\eta }$. Such a property is satisfied by almost all $d$-regular graphs and for even $d$, by the $(d\mathrm{/}2)$-th power of a Hamilton cycle.