Proving a directed analogue of the Gy\'arf\'as-Sumner conjecture for orientations of $P_4$

TL;DR

This paper advances the understanding of the dichromatic number in oriented graphs by proving a directed analogue of the Gyárfás-Sumner conjecture for orientations of P4, showing that certain $F$-free graphs have bounded chromatic number.

Contribution

It proves the first case of the directed Gyárfás-Sumner conjecture for orientations of P4, establishing boundedness of the dichromatic number in this setting.

Findings

Proved the conjecture for orientations of P4.

Established bounded dichromatic number for P4-free oriented graphs.

Progressed towards the general conjecture for all forests.

Abstract

An oriented graph is a digraph that does not contain a directed cycle of length two. An (oriented) graph $D$ is $H$-free if $D$ does not contain $H$ as an induced sub(di)graph. The Gy\'arf\'as-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 Gy\'arf\'as-Sumner Conjecture to Digraphs; E-JC 2021] proposed an analogue of this conjecture to the dichromatic number of oriented graphs. The 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 $\overrightarrow{\chi}$-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 $\overrightarrow{\chi}$-boundedness conjecture by showing that it holds when $F$ is any orientation of a path on four vertices.