Orientations of cycles in digraphs of high chromatic number and high minimum out-degree

Abstract

We characterize all orientations of cycles $C$ for which for every fixed $\epsilon >0$ there exists a constant $c\ge 1$ such that every digraph $D$ without loops or parallel arcs with $\chi (D)\ge c$ and minimum out-degree at least $\epsilon |V(D)|$ contains $C$ as a subdigraph. This generalizes a result of Thomassen.