Coloring Delaunay-Edges and their Generalizations

TL;DR

This paper explores advanced coloring problems in geometric hypergraphs, focusing on coloring Delaunay-edges and t-subsets to ensure diverse color distribution within hyperedges, extending classical vertex coloring concepts.

Contribution

It introduces a generalized framework for coloring t-subsets in geometric hypergraphs, including Delaunay-edges, and investigates various geometric regions and their coloring properties.

Findings

Established coloring results for Delaunay-edges with respect to various geometric regions

Connected vertex coloring and t-subset coloring in geometric and abstract hypergraphs

Extended classical coloring problems to t-subsets and Delaunay-graphs

Abstract

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 \emph{$t$-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 \emph{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.