Coloring non-crossing strings

Louis Esperet (1) , Daniel Gonçalves (2) , Arnaud Labourel (3)

1 G-SCOP_OC - Optimisation Combinatoire
2 ALGCO - Algorithmes, Graphes et Combinatoire
3 LIF - Laboratoire d'informatique Fondamentale de Marseille

For a family F of geometric objects in the plane, define χ(F) as the least integer such that the elements of F can be colored with colors, in such a way that any two intersecting objects have distinct colors. When F is a set of pseudo-disks that may only intersect on their boundaries, and such that any point of the plane is contained in at most k pseudo-disks, it can be proved that χ(F) 3k/2 + o(k) since the problem is equivalent to cyclic coloring of plane graphs. In this paper, we study the same problem when pseudo-disks are replaced by a family F of pseudo-segments (a.k.a. strings) that do not cross. In other words, any two strings of F are only allowed to " touch " each other. Such a family is said to be k-touching if no point of the plane is contained in more than k elements of F. We give bounds on χ(F) as a function of k, and in particular we show that k-touching segments can be colored with k + 5 colors. This partially answers a question of Hliněn´Hliněn´y (1998) on the chromatic number of contact systems of strings. * A preliminary version of this work appeared in the proceedings of EuroComb'09 [5].

Mots clés: Graph coloring; String graphs

Domaines: Calcul parallèle, distribué et partagé [cs.DC]

Louis Esperet, Daniel Gonçalves, Arnaud Labourel. Coloring non-crossing strings. The Electronic Journal of Combinatorics, 2016, 23 (4), pp.4.4. ⟨10.37236/5710⟩. ⟨hal-01480244⟩