Subdivisions of oriented cycles in digraphs with large chromatic number

Abstract : Extended 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).
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Bordeaux Graph Wokshop 2016, Nov 2016, Bordeaux, France. Proceedings of BGW 216 (Bordeaux Graph Wokshop 2016), pp.85-88, <http://bgw.labri.fr/2016/>
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Nathann Cohen, Frédéric Havet, William Lochet, Nicolas Nisse. Subdivisions of oriented cycles in digraphs with large chromatic number. Bordeaux Graph Wokshop 2016, Nov 2016, Bordeaux, France. Proceedings of BGW 216 (Bordeaux Graph Wokshop 2016), pp.85-88, <http://bgw.labri.fr/2016/>. <hal-01411115>

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