Spanning trees in dense directed graphs

Abstract

In 2001, Komlós, Sárközy and Szemerédi proved that, for each α>0, there is some c>0 and n0 such that, if n≥n0, then every n-vertex graph with minimum degree at least (1/2+α)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 α>0, there is some c>0 and n0 such that, if n≥n0, then every n-vertex directed graph with minimum semi-degree at least (1/2+α)n contains a copy of every n-vertex oriented tree whose underlying maximum degree is at most cn/log⁡n.As with Komlós, Sárközy and Szemerédi'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 Δ, for any constant Δ∈N and sufficiently large n. In contrast to these results, our methods do not use Szemerédi's regularity lemma.