We consider geometric hypergraphs whose vertex set is a finite set of points (e.g., in the plane), and whose hyperedges are the intersections of this set with a family of geometric regions (e.g., axis-parallel rectangles). A typical coloring problem for such geometric hypergraphs asks, given an integer k, for the existence of an integer m=m(k), such that every set of points can be k-colored such that every hyperedge of size at least m contains points of different (or all k) colors. We generalize this notion by introducing coloring of t-subsets of points such that every hyperedge that contains enough points contains t-subsets of different (or all) colors. In particular, we consider all t-subsets and t-subsets that are themselves hyperedges. The latter, with t=2, is equivalent to coloring the edges of the so-called Delaunay-graph. In this paper we study colorings of Delaunay-edges with respect to halfplanes, pseudo-disks, axis-parallel and bottomless rectangles, and also discuss colorings of t-subsets of geometric and abstract hypergraphs, and connections between the standard coloring of vertices and coloring of t-subsets of vertices.
Bibliographical noteFunding Information:
Research by the first author was partially supported by ERC AdG no 267165 (Disconv) and MTA EU10/2016-11001 . Research by the second author was supported by the National Research, Development and Innovation Office – NKFIH under the grant K 116769 and K 132696 . Research by the second and third authors was supported by the Lendület program of the Hungarian Academy of Sciences (MTA), under grant number LP2017-19/2017 .
© 2021 Elsevier B.V.
- Geometric hypergraphs
ASJC Scopus subject areas
- Computer Science Applications
- Geometry and Topology
- Control and Optimization
- Computational Theory and Mathematics
- Computational Mathematics