Proving a directed analogue of the Gyárfás-Sumner conjecture for orientations of \(P_4\)

DOI10.37236/11538arXiv2209.06171OpenAlexW4386916663WikidataQ123001489 ScholiaQ123001489MaRDI QIDQ6199196

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Publication date: 23 February 2024

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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 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 $overrightarrowchi$-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 $overrightarrowchi$-boundedness conjecture by showing that it holds when $F$ is any orientation of a path on four vertices.

Full work available at URL: https://arxiv.org/abs/2209.06171

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