Coloring Delaunay-Edges and their Generalizations
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We consider geometric graphs 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 (unordered) t-tuples of points such that every hyperedge that contains enough points contains t-tuples of different (or all) colors. In particular, we consider all t-tuples and t-tuples 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-tuples of geometric and abstract hypergraphs, and connections between the standard coloring of vertices and coloring of t-tuples of vertices.
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