Semidegree threshold for spanning trees in oriented graphs

TL;DR

This paper proves that oriented graphs with sufficiently high semidegree contain all bounded-degree oriented trees, establishing an asymptotically optimal threshold related to the semidegree for such spanning trees.

Contribution

It establishes a sharp threshold for the minimum semidegree ensuring the presence of all bounded-degree oriented spanning trees in large oriented graphs.

Findings

Threshold of (3/8 + γ)n for semidegree guarantees spanning trees

Asymptotically optimal result for oriented tree containment

Applicable to all bounded-degree oriented trees on large graphs

Abstract

We show that for all $\gamma > 0$ and $\Delta \in \mathbb{N}$, there is some $n_0$ such that, if $n \geq n_0$, then every oriented graph on $n$ vertices with minimum semidegree at least $(3/8 + \gamma)n$ contains a copy of each oriented tree on $n$ vertices with maximum degree at most $\Delta$. This is asymptotically best possible.