Coloring Intersection Hypergraphs of Pseudo-Disks

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.

On the number of hyperedges in the hypergraph of lines and pseudo-discs

“…Furthermore, it was proved in [4] that for any fixed t, the number of hyperedges of H(P, F) of size t is bounded by O(t 2 |P |). This result was also generalized in [8] (see also [2]) to the case where P is a family of pseudo-discs.…”

“…In particular, it was proved in [4] that the Delaunay graph of H(P, F) (namely, the restriction of H to hyperedges of size 2) is planar. This result was generalized in [8] to the case where P is a family of pseudo-discs instead of points, and the hyperedges are defined by non-empty intersections of any element in F with the elements of P . Furthermore, it was proved in [4] that for any fixed t, the number of hyperedges of H(P, F) of size t is bounded by O(t 2 |P |).…”

Strong Hanani-Tutte for the Torus

“…Keszegh [18] gave a more uniform treatment of these types of results based on hypergraphs, which at the core again uses the Hanani-Tutte theorem. While he only states results for the plane, all of his material should lift to the projective plane and the torus based on the availability of the Hanani-Tutte theorem on those surfaces.…”

The $ε$-$t$-Net Problem

“…The first one is about an analogue of the Delaunay graph for non-piercing regions [37]. The important specific case where the regions are pseudo-disks had already been studied [5,25,26]. Definition 25.…”