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 \(d=2\) .
