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 $\mathcal C$, a digraph $H$ is a hero in $\mc C$ if $H$-free digraphs of $\mathcal C$ 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.

翻译：有向图的双色数是指将其顶点划分为无环诱导子图的最小规模。给定有向图类$\mathcal C$，若有向图$H$满足$\mathcal C$中所有不含$H$作为诱导子图的有向图均具有有界双色数，则称$H$为$\mathcal C$中的英雄。在一篇开创性论文中，Berger等人给出了锦标赛中所有英雄的简单刻画。本文证明了拟传递有向图中的英雄与锦标赛中的英雄相同，并给出了简洁证明。此外，我们证明了在多部有向图类中这一结论不成立，从而否定了Aboulker、Charbit和Naserasr提出的猜想。最后，我们给出了定向完全多部图中英雄的完整刻画，仅需解决一个6顶点锦标赛的特殊情况。