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

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 $\epsilon \mathrm{\mid }V(D)\mathrm{\mid }$ contains $C$ as a subdigraph. This generalizes a result of Thomassen.