Bounded-degree spanning trees in randomly perturbed graphs

Abstract

We show that for any fixed dense graph G and bounded-degree tree T on the same number of vertices, a modest random perturbation of G will typically contain a copy of T . This combines the viewpoints of the well-studied problems of embedding trees into fixed dense graphs and into random graphs, and extends a sizeable body of existing research on randomly perturbed graphs. Specifically, we show that there is $c = c(α,Δ)$ such that if G is an n-vertex graph with minimum degree at least $αn$, and T is an n-vertex tree with maximum degree at most $Δ$ , then if we add cn uniformly random edges to G, the resulting graph will contain T asymptotically almost surely (as $n\to\infty$ ). Our proof uses a lemma concerning the decomposition of a dense graph into super-regular pairs of comparable sizes, which may be of independent interest.