SUMMARY
For a family of geometric objects in the plane F={S1,…,Sn}\mathcal{F}=\{S_1,\ldots,S_n\}, define ?(F)\chi(\mathcal{F}) as the least integer l\ell such that the elements of F\mathcal{F} can be colored with l\ell colors, in such a way that any two intersecting objects have distinct colors. When F\mathcal{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 kk pseudo-disks, it can be proven that ?(F)=3k/2+o(k)\chi(\mathcal{F})\le 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\mathcal{F} of pseudo-segments (a.k.a. strings) that do not cross. In other words, any two strings of F\mathcal{F} are only allowed to "touch" each other. Such a family is said to be kk-touching if no point of the plane is contained in more than kk elements of F\mathcal{F}. We give bounds on ?(F)\chi(\mathcal{F}) as a function of kk, and in particular we show that kk-touching segments can be colored with k+5k+5 colors. This partially answers a question of Hlinený (1998) on the chromatic number of contact systems of strings.