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arXiv · 1605.07762

Subdivisions of oriented cycles in digraphs with large chromatic number

Nathann Cohen·Frédéric Havet·William Lochet·Nicolas Nisse
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Abstract

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 $C$ a cycle with two blocks, every strongly connected digraph with sufficiently large chromatic number contains a subdivision of $C$. We prove a similar result 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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Nathann Cohen, Frédéric Havet, William Lochet, Nicolas Nisse. 2016-05-25. Subdivisions of oriented cycles in digraphs with large chromatic number. https://arxiv.org/abs/1605.07762

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