Coloring Dense Digraphs


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 Erdős-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 α(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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Correspondence to Ararat Harutyunyan.

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Harutyunyan, A., Le, TN., Newman, A. et al. Coloring Dense Digraphs. Combinatorica 39, 1021–1053 (2019).

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Mathematics Subject Classification (2010)

  • 05C15