Oriented trees in digraphs with large girth

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 $T$ with $k$ edges whose maximum degree does not exceed the minimum degree of $G$. Let $\delta^0(D)$ be the minimum semidegree of a digraph $D$ and $\Delta(D)$ be the maximum degree of $D$. In this paper, we establish a digraph version of Jiang's result, stating that every oriented graph $D$ of girth at least $2\ell+1$ with $\delta^0(D)\ge \max\{k/\ell,\Delta(T)\}$ contains every oriented tree with $k$ edges, that answers a question raised by Stein and Trujillo-Negrete in affirmative.