The exact chromatic number of the convex segment disjointness graph

Given a set of n points in the plane in general and convex position, let n be the set of closed line segments joining pairs of elements in the point set. Let Dn be the graph whose vertex set is n, where two line segments are adjacent if and only if they are disjoint. In a more general setting, Araujo, Dumitrescu, Hurtado, Noy, and Urrutia introduced the problem of determining the chromatic number of Dn. Fabila-Monroy and Wood showed that a lower bound is given by n − ⌊ q 2n + 1 − 1 ⌋. The main result of the present note is that the chromatic number actually equals this lower bound. The proof is constructive.