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- W4380876845 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." @default.
- W4380876845 created "2023-06-17" @default.
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- W4380876845 date "2023-01-01" @default.
- W4380876845 modified "2023-10-14" @default.
- W4380876845 title "Proving a directed analogue of the Gyárfás-Sumner conjecture for orientations of P4" @default.
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- W4380876845 doi "https://doi.org/10.5817/cz.muni.eurocomb23-042" @default.
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