Coloring Intersection Hypergraphs of Pseudo-Disks
Balázs Keszegh · Discrete & Computational Geometry · 2019
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 four colors and a conflict-free coloring with $$O(\log n)$$ 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 $$O(\log n)$$ 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.