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