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arXiv · 2209.06171

Proving a directed analogue of the Gyárfás-Sumner conjecture for orientations of $P_4$

Linda Cook·Tomáš Masařík·Marcin Pilipczuk·Amadeus Reinald·Uéverton S. Souza
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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 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 $ω(G)$ has chromatic number at most $f(ω(G))$. 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 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 $\overrightarrowχ$-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 $f(ω(D))$, where $ω(D)$ 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 $\overrightarrowχ$-boundedness conjecture by showing that it holds when $F$ is any orientation of a path on four vertices.

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Linda Cook, Tomáš Masařík, Marcin Pilipczuk, Amadeus Reinald, Uéverton S. Souza. 2022-09-13. Proving a directed analogue of the Gyárfás-Sumner conjecture for orientations of $P_4$. https://doi.org/10.37236/11538%2010.5817%2Fcz.muni.eurocomb23-042

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