Heroes in oriented complete multipartite graphs

Abstract

The dichromatic number of a digraph is the minimum size of a partition of its vertices into acyclic induced subgraphs. Given a class of digraphs C, a digraph H is a hero in C if H-free digraphs of C have bounded dichromatic number. In a seminal paper, Berger at al. give a simple characterization of all heroes in tournaments. In this paper, we give a simple proof that heroes in quasi-transitive oriented graphs are the same as heroes in tournaments. We also prove that it is not the case in the class of oriented multipartite graphs, disproving a conjecture of Aboulker, Charbit and Naserasr. We also give a full characterisation of heroes in oriented complete multipartite graphs up to the status of a single tournament on 6 vertices.

Proving a Directed Analogue of the Gyárfás-Sumner Conjecture for Orientations of $P_4$

“…It is easy to see that every heroic set must forbid some digraph with a digon and that every heroic set must forbid some oriented graph. However, we are far from a characterization such as the one in [6] despite significant work towards better understanding heroic sets (such as [1,3,5,9]). It was shown in [5] that: For every oriented forest F , if the set consisting of a digon, F , and a digraph H is heroic, then F is the disjoint union of oriented stars or H is a transitive tournament.…”