Résumé
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, 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 $d=2$.