Spanning trees in randomly perturbed graphs

Abstract

A classical result of Komlós, Sárközy and Szemerédi states that every $n$-vertex graph with minimum degree at least $(1/2+ o(1))n$ contains every $n$-vertex tree with maximum degree $O(n/\log{n})$ as a subgraph, and the bounds on the degree conditions are sharp. On the other hand, Krivelevich, Kwan and Sudakov recently proved that for every $n$-vertex graph $G_α$ with minimum degree at least $αn$ for any fixed $α>0$ and every $n$-vertex tree $T$ with bounded maximum degree, one can still find a copy of $T$ in $G_α$ with high probability after adding $O(n)$ randomly-chosen edges to $G_α$. We extend their results to trees with unbounded maximum degree. More precisely, for a given $n^{o(1)}\leq Δ\leq cn/\log n$ and $α>0$, we determine the precise number (up to a constant factor) of random edges that we need to add to an arbitrary $n$-vertex graph $G_α$ with minimum degree $αn$ in order to guarantee a copy of any fixed $n$-vertex tree $T$ with maximum degree at most~$Δ$ with high probability.

41 pages