Aboulker P.Bang-Jensen J.Bousquet N.Charbit P.Havet F.Maffray F.Zamora J.2022-07-022022-07-022018-11Journal of Graph Theory Volume 89, Issue 3, Pages 304 - 326November 201803649024https://repositorio.unab.cl/xmlui/handle/ria/23129Indexación ScopusA famous conjecture of Gyárfás and Sumner states for any tree T and integer k, if the chromatic number of a graph is large enough, either the graph contains a clique of size k or it contains T as an induced subgraph. We discuss some results and open problems about extensions of this conjecture to oriented graphs. We conjecture that for every oriented star S and integer k, if the chromatic number of a digraph is large enough, either the digraph contains a clique of size k or it contains S as an induced subgraph. As an evidence, we prove that for any oriented star S, every oriented graph with sufficiently large chromatic number contains either a transitive tournament of order 3 or S as an induced subdigraph. We then study for which sets p of orientations of P4 (the path on four vertices) similar statements hold. We establish some positive and negative results. © 2018 Wiley Periodicals, Inc.enGraphChromatic NumberCliqueχ-boundedArtículo10.1002/jgt.22252