The χ -Binding Function of d-Directional Segment Graphs.

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 $R^2$ 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 $\chi \left(G\right)$ 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, Dvořák 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 Nešetřil, which treated the special case $d=2$ .