Coloring Jordan Regions and Curves

A pseudo-disk is a subset of the plane that is homeomorphic to a closed disk. Consider a family F of pseudo-disks whose interiors are pairwise disjoint, and such that any two pseudo-disks intersect in at most one point. If any point of the plane is contained in at most k elements of F (with k sufficiently large), then we show that the elements of F can be colored with at most k + 1 colors so that intersecting pseudo-disks are assigned distinct colors. This is best possible and answers a question raised by Reed and Shepherd in 1996. As a simple corollary, we also obtain a positive answer to a problem of Hliněný (1998) on the chromatic number of contact systems of strings. We also investigate the chromatic number of families of touching Jordan curves. This can be used to bound the ratio between the maximum number of vertex-disjoint directed cycles in a planar digraph, and its fractional counterpart.