χ‐bounded families of oriented graphs

• Published in: Journal of graph theory Vol. 89; no. 3; pp. 304 - 326
• Main Authors: Aboulker, P., Bang‐Jensen, J., Bousquet, N., Charbit, P., Havet, F., Maffray, F., Zamora, J.
• Format: Journal Article
• Language: English
• Published: Hoboken Wiley Subscription Services, Inc 01.11.2018; Wiley
• Subjects: chromatic number; clique number; Computer Science; Discrete Mathematics; Graph theory; Graphs; χ‐bounded
• ISSN: 0364-9024; 1097-0118
• DOI: 10.1002/jgt.22252

A 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.