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

TL;DR

This paper investigates the chromatic number bounds of intersection graphs formed by line segments with limited slopes, establishing exact bounds and confirming a conjecture for even clique numbers.

Contribution

It determines the exact $ ext{chi}$-binding function for $d$-directional segment graphs, extending previous results and confirming a conjecture for even clique numbers.

Findings

The $ ext{chi}$-binding function is $d ext{omega}$ for even $ ext{omega}$.

Constructed graphs meet the bound exactly for even $ ext{omega}$.

Extended the known bounds from $d=2$ to general $d$.

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 $d=2$.