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arXiv · 1704.07219

Coloring dense digraphs

Ararat Harutyunyan·Tien-Nam Le·Alantha Newman·Stéphan Thomassé
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Abstract

The chromatic number of a digraph $D$ is the minimum number of acyclic subgraphs covering the vertex set of $D$. A tournament $H$ is a hero if every $H$-free tournament $T$ has chromatic number bounded by a function of $H$. Inspired by the celebrated Erd\H{o}s--Hajnal conjecture, Berger et al. fully characterized the class of heroes in 2013. We extend this framework to dense digraphs: A digraph $H$ is a superhero if every $H$-free digraph $D$ has chromatic number bounded by a function of $H$ and $\alpha(D)$, the independence number of the underlying graph of $D$. We prove here that a digraph is a superhero if and only if it is a hero, and hence characterize all superheroes. This answers a question of Aboulker, Charbit and Naserasr.

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Ararat Harutyunyan, Tien-Nam Le, Alantha Newman, Stéphan Thomassé. 2017-04-24. Coloring dense digraphs. https://arxiv.org/abs/1704.07219

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