Subdivisions of oriented cycles in digraphs with large chromatic number

International audienceExtended Abstract The chromatic number χ(D) of a digraph D is the chromatic number of its underlying graph. An oriented cycle is an orientation of a undirected cycle. We first show that for any oriented cycle C, there are digraphs containing no subdivision of C (as a subdigraph) and arbitrarily large chromatic number. In contrast, we show that for any cycle with two blocks C, every strongly connected digraph with sufficiently large chromatic number contains a subdivision of C. More generally, we conjecture that this result holds for any oriented cycle. As a further evidence, we prove this conjecture for the antidirected cycle on four vertices (in which two vertices have out-degree 2 and two vertices have in-degree 2)