On The Chromatic Number of Geometric Hypergraphs

• Authors: Shakhar Smorodinsky
• Venue: SIAM Journal on Discrete Mathematics
• Year: 2007

A finite family $\mathcal{R}$ of simple Jordan regions in the plane defines a hypergraph $H=H(\mathcal{R})$ where the vertex set of $H$ is $\mathcal{R}$ and the hyperedges are all subsets $S \subset \R$ for which there is a point $p$ such that $S = \{r \in \mathcal{R} | p \in r\}$. The chromatic number of $H(\mathcal{R})$ is the minimum number of colors needed to color the members of $\mathcal{R}$ such that no hyperedge is monochromatic. In this paper we initiate the study of the chromatic number of such hypergraphs and obtain the following results: (i) Any hypergraph that is induced by a family of $n$ simple Jordan regions such that the maximum union complexity of any $k$ of them (for $1\leq k \leq m$) is bounded by $\mathcal{U}(m)$ and $\frac{\mathcal{U}(m)}{m}$ is a nondecreasing function is $O(\frac{\mathcal{U}(n)}{n})$-colorable. Thus, for example, we prove that any finite family of pseudo-discs can be colored with a constant number of colors. (ii) Any hypergraph induced by a finite family of planar discs is four colorable. This bound is tight. In fact, we prove that this statement is equivalent to the four-color theorem. (iii) Any hypergraph induced by $n$ axis-parallel rectangles is $O(\log n)$-colorable. This bound is asymptotically tight. Our proofs are constructive. Namely, we provide deterministic polynomial-time algorithms for coloring such hypergraphs with only “few” colors (that is, the number of colors used by these algorithms is upper bounded by the same bounds we obtain on the chromatic number of the given hypergraphs). As an application of (i) and (ii) we obtain simple constructive proofs for the following: (iv) Any set of $n$ Jordan regions with near linear union complexity admits a conflict-free (CF) coloring with polylogarithmic number of colors. (v) Any set of $n$ axis-parallel rectangles admits a CF-coloring with $O(\log^2(n))$ colors.