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arXiv · 2309.06072

The $\chi$-binding function of $d$-directional segment graphs

Lech Duraj·Ross J. Kang·Hoang La·Jonathan Narboni·Filip Pokrývka·Clément Rambaud·Amadeus Reinald
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Abstract

Given a positive integer $d$, the class $d$-DIR is defined as all those intersection graphs formed from a finite collection of line segments in ${\mathbb R}^2$ having at most $d$ slopes. Since each slope induces an interval graph, it easily follows for every $G$ in $d$-DIR with clique number at most $\omega$ that the chromatic number $\chi(G)$ of $G$ is at most $d\omega$. We show for every even value of $\omega$ how to construct a graph in $d$-DIR that meets this bound exactly. This partially confirms a conjecture of Bhattacharya, Dvo\v{r}\'ak and Noorizadeh. Furthermore, we show that the $\chi$-binding function of $d$-DIR is $\omega \mapsto d\omega$ for $\omega$ even and $\omega \mapsto d(\omega-1)+1$ for $\omega$ odd. This extends an earlier result by Kostochka and Ne\v{s}et\v{r}il, which treated the special case $d=2$.

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Lech Duraj, Ross J. Kang, Hoang La, Jonathan Narboni, Filip Pokrývka, Clément Rambaud, Amadeus Reinald. 2023-09-12. The $\chi$-binding function of $d$-directional segment graphs. https://arxiv.org/abs/2309.06072

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