Bose, Prosenjit and Cardinal, Jean and Collette, Sébastien and Hurtado, Ferran and Korman, Matiaset al. - Coloring and Guarding Arrangements

dmtcs:2072 - Discrete Mathematics & Theoretical Computer Science, December 4, 2013, Vol. 15 no. 3
Coloring and Guarding Arrangements

Authors: Bose, Prosenjit and Cardinal, Jean and Collette, Sébastien and Hurtado, Ferran and Korman, Matias and Langerman, Stefan and Taslakian, Perouz

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 Ω(logn/loglogn). and O(n√). Similarly, we give bounds on the minimum size of a subset S of the intersections of the lines in 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.

Volume: Vol. 15 no. 3
Section: Combinatorics
Published on: December 4, 2013
Submitted on: February 22, 2012
Keywords: line arrangement,hypergraph coloring,duality,independent set,vertex cover,[INFO.INFO-DM] Computer Science [cs]/Discrete Mathematics [cs.DM]