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An oriented cycle is an orientation of a undirected cycle. We first show thatfor any oriented cycle $C$, there are digraphs containing no subdivision of $C$(as a subdigraph) and arbitrarily large chromatic number. In contrast, we showthat for any $C$ a cycle with two blocks, every strongly connected digraph withsufficiently large chromatic number contains a subdivision of $C$. We prove asimilar result for the antidirected cycle on four vertices (in which twovertices have out-degree $2$ and two vertices have in-degree $2$).

Subdivisions of oriented cycles in digraphs with large chromatic number