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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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https://hal.archives-ouvertes.fr/hal-01411115
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Submitted on : Wednesday, December 7, 2016 - 9:40:41 AM
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  • HAL Id : hal-01411115, version 1

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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. pp.85-88. ⟨hal-01411115⟩

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