χ-bounded families of oriented graphs

Pierre Aboulker (1) , Jørgen Bang-Jensen (2) , Nicolas Bousquet (3) , Pierre Charbit (4, 5) , Frédéric Havet (1) , Frédéric Maffray (3) , Jose Zamora (6)

1 COATI - Combinatorics, Optimization and Algorithms for Telecommunications
2 SDU - University of Southern Denmark
3 G-SCOP_OC - Optimisation Combinatoire
4 IRIF (UMR_8243) - Institut de Recherche en Informatique Fondamentale
5 GANG - Networks, Graphs and Algorithms
6 UNAB - Universidad Andrés Bello [Santiago]

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 $P_4$ (the path on four vertices) similar statements hold. We establish some positive and negative results.

Domaines: Mathématique discrète [cs.DM]

Pierre Aboulker, Jørgen Bang-Jensen, Nicolas Bousquet, Pierre Charbit, Frédéric Havet, et al.. χ-bounded families of oriented graphs. Journal of Graph Theory, 2018, 89 (3), pp.304 - 326. ⟨10.1002/jgt.22252⟩. ⟨hal-01882395⟩