Coloring Non-Crossing Strings

For a family of geometric objects in the plane F = {S1, . . . , Sn}, 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 proven 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 pseudodisks 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 Hliněný (1998) on the chromatic number of contact systems of strings. E-mail addresses: [email protected], [email protected], [email protected]. A preliminary version of this work appeared in the proceedings of EuroComb’09 [5]. Louis Esperet is partially supported by ANR Project Stint (anr-13-bs02-0007) and LabEx PERSYVAL-Lab (anr-11-labx-0025-01). Arnaud Labourel is partially supported by ANR project MACARON (anr-13-js02-0002). 1 ar X iv :1 51 1. 03 82 7v 1 [ m at h. C O ] 1 2 N ov 2 01 5 2 L. ESPERET, D. GONÇALVES, AND A. LABOUREL