Coloring Planar Homothets and Three-Dimensional Hypergraphs

The inclusion relation between simple objects in the plane may be used to define geometric set systems, or hypergraphs. Properties of various types of colorings of these hypergraphs have been the subject of recent investigations, with applications to wireless networking. We first prove that every set of homothetic copies of a given convex body in the plane can be colored with four colors so that any point covered by at least two copies is covered by two copies with distinct colors. This generalizes a previous result from Smorodinsky [18]. As a corollary, we find improvements to well studied variations of the coloring problem such as conflict-free colorings, k-strong (conflict-free) colorings and choosability. We also show a relation between our proof and Schnyder's characterization of planar graphs. Then we show that for any k >1, every three-dimensional hypergraph can be colored with 6(k - 1) colors so that every hyperedge e contains min{|e|, k} vertices with mutually distinct colors. Furthermore, we also show that at least 2k colors might be necessary. This refines a previous result from Aloupis et al. [2].