Coloring and Guarding Arrangements

TL;DR

This paper investigates the minimum number of colors needed to color lines in a plane so that no cell is monochromatic, providing bounds and reformulating the problem as hypergraph coloring and guarding problems.

Contribution

It introduces new bounds on the chromatic number for line arrangements and models the problem as hypergraph coloring and vertex cover variants.

Findings

Chromatic number is between Ω(log n / log log n) and O(√n).

Bounds on minimum guarding sets for cells in line arrangements.

Reformulation as hypergraph coloring and vertex cover problems.

Abstract

Given an arrangement of lines in the plane, what is the minimum number $c$ of colors required to color the lines so that no cell of the arrangement is monochromatic? In this paper we give bounds on the number c both for the above question, as well as some of its variations. We redefine these problems as geometric hypergraph coloring problems. If we define $\Hlinecell$ as the hypergraph where vertices are lines and edges represent cells of the arrangement, the answer to the above question is equal to the chromatic number of this hypergraph. We prove that this chromatic number is between $\Omega (\log n / \log\log n)$. and $O(\sqrt{n})$. Similarly, we give bounds on the minimum size of a subset $S$ of the intersections of the lines in $\mathcal{A}$ such that every cell is bounded by at least one of the vertices in $S$. This may be seen as a problem on guarding cells with vertices when the lines act as obstacles. The problem can also be defined as the minimum vertex cover problem in the hypergraph $\Hvertexcell$, the vertices of which are the line intersections, and the hyperedges are vertices of a cell. Analogously, we consider the problem of touching the lines with a minimum subset of the cells of the arrangement, which we identify as the minimum vertex cover problem in the $\Hcellzone$ hypergraph.