Randomly perturbed digraphs also have bounded-degree spanning trees

Abstract

We show that a randomly perturbed digraph, where we start with a dense digraph $D_{\alpha }$ and add a small number of random edges to it, will typically contain a fixed orientation of a bounded degree spanning tree. This answers a question posed by Araujo, Balogh, Krueger, Piga and Treglown and generalizes the corresponding result for randomly perturbed graphs by Krivelevich, Kwan and Sudakov. More specifically, we prove that there exists a constant $c=c(\alpha ,\Delta )$ such that if $T$ is an oriented tree with maximum degree $\Delta$ and $D_{\alpha }$ is an $n$-vertex digraph with minimum semidegree $\alpha n$, then the graph obtained by adding $cn$ uniformly random edges to $D_{\alpha }$ will contain $T$ with high probability.

arXiv comment10 pages