Coloring Jordan regions and curves

TL;DR

This paper proves that families of Jordan regions with limited intersections can be colored with a number of colors close to the maximum overlap, solving a longstanding problem and applying results to planar digraphs.

Contribution

It establishes optimal coloring bounds for Jordan regions with limited intersections, answering a question from 1996 and extending to contact systems of strings and planar digraphs.

Findings

Coloring of Jordan regions with maximum intersection number k requires at most k+1 colors.

The result is optimal, matching known lower bounds.

Applications to contact systems of strings and bounds on directed cycles in planar digraphs.

Abstract

A Jordan region is a subset of the plane that is homeomorphic to a closed disk. Consider a family $\mathcal{F}$ of Jordan regions whose interiors are pairwise disjoint, and such that any two Jordan regions intersect in at most one point. If any point of the plane is contained in at most $k$ elements of $\mathcal{F}$ (with $k$ sufficiently large), then we show that the elements of $\mathcal{F}$ can be colored with at most $k+1$ colors so that intersecting Jordan regions 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 Hlin\v{e}n\'y (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.