Coloring half-planes and bottomless rectangles

TL;DR

This paper investigates the chromatic number of hypergraphs formed by geometric regions like bottomless rectangles and half-planes, providing bounds, solutions for special cases, and efficient coloring algorithms.

Contribution

It offers the first complete solution for bottomless rectangles, near-complete results for half-planes, and introduces efficient coloring algorithms for these geometric hypergraphs.

Findings

Complete solution for bottomless rectangles

Near-complete results for half-planes

Efficient coloring algorithms provided

Abstract

We prove lower and upper bounds for the chromatic number of certain hypergraphs defined by geometric regions. This problem has close relations to conflict-free colorings. One of the most interesting type of regions to consider for this problem is that of the axis-parallel rectangles. We completely solve the problem for a special case of them, for bottomless rectangles. We also give an almost complete answer for half-planes and pose several open problems. Moreover we give efficient coloring algorithms.