Proving a directed analogue of the Gy\'arf\'as-Sumner conjecture for orientations of $P_4$
| Autor: | Cook, Linda, Masařík, Tomáš, Pilipczuk, Marcin, Reinald, Amadeus, Souza, Uéverton S. |
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| Rok vydání: | 2022 |
| Předmět: | |
| Zdroj: | The Electronic Journal of Combinatorics, 30(3), 36:1-36:27, 2023; Proceedings: European Conference on Combinatorics, Graph Theory and Applications, EUROCOMB 2023 |
| Druh dokumentu: | Working Paper |
| DOI: | 10.37236/11538 |
| Popis: | 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. Comment: 24 pages, 5 figures |
| Databáze: | arXiv |
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