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arXiv · 2306.14648

Randomly perturbed digraphs also have bounded-degree spanning trees

Patryk Morawski·Kalina Petrova
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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.

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Patryk Morawski, Kalina Petrova. 2023-06-26. Randomly perturbed digraphs also have bounded-degree spanning trees. https://arxiv.org/abs/2306.14648

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