Coloring a set of touching strings

For a family of geometric objects in the plane F = {S1, . . . , Sn}, define χ(F) as the least integer l such that the elements of F can be colored with l colors, in such a way that any two intersecting objects have distinct colors. When F is a set of Jordan regions that may only intersect on their boundaries, and such that any point of the plane is contained in at most k regions, 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 Jordan regions are replaced by a family F of Jordan curves that do not cross. In other words, any two curves of F are only allowed to “touch” each other. We conjecture that also in this case, χ(F) only depends on the maximum number of curves containing a given point of the plane. To support this conjecture, we prove it when the curves are x-monotone (any vertical line intersect each curve in at most one point), and we give a bound in the general case that also depends on how many times two curves intersect.