Oriented Trees in Digraphs without Oriented $4$-cycles

TL;DR

This paper proves that certain digraphs without oriented 4-cycles contain all oriented trees with a specific number of arcs, extending understanding of tree embeddings in constrained digraphs.

Contribution

It establishes conditions under which digraphs without oriented 4-cycles contain all oriented trees with a given number of arcs, improving results for special tree types.

Findings

Digraphs with high minimum semidegree contain all oriented trees with k arcs.

No oriented 4-cycle condition guarantees tree embeddings.

Improved results for antidirected and arborescence trees.

Abstract

We prove that if $D$ is a digraph of maximum outdegree and indegree at least $k$, and minimum semidegree at least $k/2$ that contains no oriented $4$-cycles, then $D$ contains each oriented tree $T$ with~$k$ arcs. This can be slightly improved if $T$ is either antidirected or an arborescence.