Randomly perturbed digraphs also have bounded-degree spanning trees

TL;DR

This paper proves that adding a small number of random edges to a dense directed graph guarantees the presence of any fixed bounded-degree oriented spanning tree, extending known results from undirected to directed graphs.

Contribution

It establishes that randomly perturbed dense digraphs almost surely contain any fixed bounded-degree oriented spanning tree, answering an open question and generalizing previous undirected graph results.

Findings

Random perturbation ensures spanning trees with high probability.

Bounded-degree oriented trees are contained in perturbed dense digraphs.

The result applies to graphs with linear minimum semidegree.

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.