Heroes in oriented complete multipartite graphs

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 ${\mathscr{C}}$, a digraph H $H$ is a hero in C ${\mathscr{C}}$ if H $H$-free digraphs of C ${\mathscr{C}}$ have bounded dichromatic number. In a seminal paper, Berger et al. give a simple characterisation of all heroes in tournaments. In this paper, we give a simple proof that heroes in quasitransitive oriented graphs (that are digraphs with no induced directed path on three vertices) 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, and give a characterisation of heroes in oriented complete multipartite graphs up to the status of a single tournament on six vertices.