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 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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References
P. Aboulker, P. Charbit and R. Naserasr: private communication.
E. Berger, K. Choromanski, M. Chudnovsky, J. Fox, M. Loebl, A. Scott, P. Seymour and S. Thomassé: Tournaments and colouring, Journal of Combinatorial Theory, Series B103 (2013), 1–20.
D. Bokal, G. Fijavž, M. Juvan, P. M. Kayll and B. Mohar: The circular chromatic number of a digraph, Journal of Graph Theory46 (2004), 227–240.
K. Choromanski, M. Chudnovsky and P. Seymour: Tournaments with nearlinear transitive subsets, Journal of Combinatorial Theory, Series B109 (2014), 228–249.
M. Chudnovsky: The Erdős-Hajnal Conjecture - A Survey, Journal of Graph Theory75 (2014), 178–190.
M. Chudnovsky, R. Kim, C.-H. Liu, P. Seymour and S. Thomassé: Domination in tournaments, Journal of Combinatorial Theory, Series B130 (2018), 98–113.
A. Harutyunyan, T.-N. Le, S. Thomassé and H. Wu: Coloring tournaments: from local to global, arXiv preprint: arXiv:1702.01607.
A. Harutyunyan and C. McDiarmid: Large Acyclic sets in oriented graphs, unpublished note.
A. Harutyunyan and B. Mohar: Strengthened Brooks Theorem for digraphs of girth three, Electronic Journal of Combinatorics18 (2011).
A. Harutyunyan and B. Mohar: Gallai's Theorem for List Coloring of Digraphs, SIAM Journal on Discrete Mathematics25(1) (2011), 170–180.
A. Harutyunyan and B. Mohar: Two results on the digraph chromatic number, Discrete Mathematics312(10) (2012), 1823–1826.
P. Keevash, Z. Li, B. Mohar and B. Reed: Digraph girth via chromatic number, SIAM Journal on Discrete Mathematics27(2) (2013), 693–696.
V. Neumann-Lara: The dichromatic number of a digraph, Journal of Combinatorial Theory, Series B33 (1982), 265–270.
V. Stearns: The voting problem, American Mathematical Monthly66 (1959), 761–763.
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Harutyunyan, A., Le, TN., Newman, A. et al. Coloring Dense Digraphs. Combinatorica 39, 1021–1053 (2019). https://doi.org/10.1007/s00493-019-3815-8
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DOI: https://doi.org/10.1007/s00493-019-3815-8