Spanning trees in dense directed graphs

TL;DR

This paper extends a classical undirected graph spanning tree result to dense directed graphs, showing that such graphs contain all large oriented trees with bounded maximum degree, without using Szemerédi's regularity lemma.

Contribution

It proves a directed analogue of a known theorem, establishing the existence of spanning oriented trees in dense directed graphs with improved methods.

Findings

Directed graphs with high semi-degree contain all large bounded-degree oriented trees.

The result is tight up to the constant factor in the degree bound.

The proof avoids Szemerédi's regularity lemma, simplifying the approach.

Abstract

In 2001, Koml\'os, S\'ark\"ozy and Szemer\'edi proved that, for each $\alpha>0$, there is some $c>0$ and $n_0$ such that, if $n\geq n_0$, then every $n$-vertex graph with minimum degree at least $(1/2+\alpha)n$ contains a copy of every $n$-vertex tree with maximum degree at most $cn/\log n$. We prove the corresponding result for directed graphs. That is, for each $\alpha>0$, there is some $c>0$ and $n_0$ such that, if $n\geq n_0$, then every $n$-vertex directed graph with minimum semi-degree at least $(1/2+\alpha)n$ contains a copy of every $n$-vertex oriented tree whose underlying maximum degree is at most $cn/\log n$. As with Koml\'os, S\'ark\"ozy and Szemer\'edi's theorem, this is tight up to the value of $c$. Our result improves a recent result of Mycroft and Naia, which requires the oriented trees to have underlying maximum degree at most $\Delta$, for any constant $\Delta\in \mathbb{N}$ and sufficiently large $n$. In contrast to these results, our methods do not use Szemer\'edi's regularity lemma.