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Subdivisions of oriented cycles in digraphs with large chromatic number

Nathann Cohen 1, 2 Frédéric Havet 2 William Lochet 2, 3 Nicolas Nisse 2
1 GALaC - LRI - Graphes, Algorithmes et Combinatoire (LRI)
LRI - Laboratoire de Recherche en Informatique
2 COATI - Combinatorics, Optimization and Algorithms for Telecommunications
CRISAM - Inria Sophia Antipolis - Méditerranée , Laboratoire I3S - COMRED - COMmunications, Réseaux, systèmes Embarqués et Distribués
3 MC2 - Modèles de calcul, Complexité, Combinatoire
LIP - Laboratoire de l'Informatique du Parallélisme
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. Subdivisions of oriented cycles in digraphs with large chromatic number. Journal of Graph Theory, Wiley, 2018, 89 (4), pp.439-456. ⟨10.1002/jgt.22360⟩. ⟨hal-01834779⟩

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