Proving a directed analogue of the Gyárfás-Sumner conjecture for orientations of P4

Abstract

An oriented graph is a digraph that does not contain a directed cycle of length two. An (oriented) graph $D$ is \emph{$H$-free} if $D$ does not contain $H$ as an induced sub(di)graph. The Gyárfás-Sumner conjecture is a widely-open conjecture on simple graphs, which states that for any forest $F$, there is some function $f$ such that every $F$-free graph $G$ with clique number $\omega(G)$ has chromatic number at most $f(\omega(G))$. Aboulker, Charbit, and Naserasr [Extension of Gyárfás-Sumner Conjecture to Digraphs; E-JC 2021] proposed an analogue of this conjecture to the dichromatic number of oriented graphs. The \emph{dichromatic number} of a digraph $D$ is the minimum number of colors required to color the vertex set of $D$ so that no directed cycle in $D$ is monochromatic. Aboulker, Charbit, and Naserasr's $\dichi$-boundedness conjecture states that for every oriented forest $F$, there is some function $f$ such that every $F$-free oriented graph $D$ has dichromatic number at most $f(\omega(D))$, where $\omega(D)$ is the size of a maximum clique in the graph underlying $D$. In this paper, we perform the first step towards proving Aboulker, Charbit, and Naserasr's $\dichi$-boundedness conjecture by showing that it holds when $F$ is any orientation of a path on four vertices.