Coloring intersection hypergraphs of pseudo-disks

TL;DR

This paper proves new coloring bounds for intersection hypergraphs of pseudo-disks, showing they can be properly colored with 4 colors and conflict-free colored with O(log n) colors, generalizing previous results.

Contribution

It introduces novel bounds for coloring intersection hypergraphs of pseudo-disks and regions with linear union complexity, extending and strengthening prior work.

Findings

Proper coloring with 4 colors for pseudo-disk intersection hypergraphs

Conflict-free coloring with O(log n) colors for the same hypergraphs

Planarity of the associated Delaunay-graph

Abstract

We prove that the intersection hypergraph of a family of $n$ pseudo-disks with respect to another family of pseudo-disks admits a proper coloring with $4$ colors and a conflict-free coloring with $O(\log n)$ colors. Along the way we prove that the respective Delaunay-graph is planar. We also prove that the intersection hypergraph of a family of $n$ regions with linear union complexity with respect to a family of pseudo-disks admits a proper coloring with constantly many colors and a conflict-free coloring with $O(\log n)$ colors. Our results serve as a common generalization and strengthening of many earlier results, including ones about proper and conflict-free coloring points with respect to pseudo-disks, coloring regions of linear union complexity with respect to points and coloring disks with respect to disks.