Oriented trees in digraphs without short non-directed cycles

Abstract

The girth of a graph $G$ is the length of a shortest cycle of $G$. Jiang (JCT-B, 2001) showed that every graph $G$ with girth at least $2\ell +1$ and minimum degree at least $k/\ell$ contains every tree with $k$ edges whose maximum degree does not exceed the minimum degree of $G$. In this paper, we extend Jiang's result to digraphs by proving that every digraph $D$ with no non-directed cycle of length between $3$ and $2\ell$ and minimum semidegree at least $k/\ell$ contains every oriented tree with $k$ edges whose maximum degree does not exceed the minimum semidegree of $D$. This answers a question raised by Stein and Trujillo-Negrete in the affirmative.