Heroes in oriented complete multipartite graphs

TL;DR

This paper characterizes heroes in various classes of oriented graphs, showing equivalences and differences, and provides a full classification in oriented complete multipartite graphs, advancing understanding of their dichromatic numbers.

Contribution

It proves heroes in quasi-transitive oriented graphs are the same as in tournaments and fully characterizes heroes in oriented complete multipartite graphs.

Findings

Heroes in quasi-transitive oriented graphs are identical to those in tournaments.

Heroes in oriented complete multipartite graphs are fully characterized.

The conjecture that heroes in oriented multipartite graphs match those in tournaments is disproved.

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 $\mathcal C$, 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.