($\overrightarrow{P_6}$, triangle)-Free Digraphs Have Bounded Dichromatic Number

Abstract

The dichromatic number of an oriented graph is the minimum size of a partition of its vertices into acyclic induced subdigraphs. We prove that oriented graphs with no induced directed path on six vertices and no triangle have bounded dichromatic number. This is one (small) step towards the general conjecture asserting that for every oriented tree $T$ and every integer $k$, any oriented graph that does not contain an induced copy of $T$ nor a clique of size $k$ has dichromatic number at most some function of $k$ and $T$ .