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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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Submitted on : Wednesday, December 7, 2016 - 9:40:41 AM
Last modification on : Tuesday, May 26, 2020 - 6:50:59 PM
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  • HAL Id : hal-01411115, version 1


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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