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 R2 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 ω that the chromatic number χ(G) of G is at most dω . We show for every even value of ω 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 χ -binding function of d-DIR is ω ↦ dω for ω even and ω ↦ d(ω - 1) + 1 for ω odd. This extends an earlier result by Kostochka and Nešetřil, which treated the special case d = 2 .