12 research outputs found
Coloring and guarding arrangements,
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, as well as some of its variations. We cast these problems as characterizing the chromatic and the independence numbers of a new family of geometric hypergraphs
General Position Subsets and Independent Hyperplanes in d-Space
Erd\H{o}s asked what is the maximum number such that every set of
points in the plane with no four on a line contains points in
general position. We consider variants of this question for -dimensional
point sets and generalize previously known bounds. In particular, we prove the
following two results for fixed :
- Every set of hyperplanes in contains a subset
of size at least , for some
constant , such that no cell of the arrangement of is bounded by
hyperplanes of only.
- Every set of points in , for some constant
, contains a subset of cohyperplanar points or points in
general position.
Two-dimensional versions of the above results were respectively proved by
Ackerman et al. [Electronic J. Combinatorics, 2014] and by Payne and Wood [SIAM
J. Discrete Math., 2013].Comment: 8 page
Covering Partial Cubes with Zones
A partial cube is a graph having an isometric embedding in a hypercube.
Partial cubes are characterized by a natural equivalence relation on the edges,
whose classes are called zones. The number of zones determines the minimal
dimension of a hypercube in which the graph can be embedded. We consider the
problem of covering the vertices of a partial cube with the minimum number of
zones. The problem admits several special cases, among which are the problem of
covering the cells of a line arrangement with a minimum number of lines, and
the problem of finding a minimum-size fibre in a bipartite poset. For several
such special cases, we give upper and lower bounds on the minimum size of a
covering by zones. We also consider the computational complexity of those
problems, and establish some hardness results
Upper and Lower Bounds on Long Dual-Paths in Line Arrangements
Given a line arrangement with lines, we show that there exists a
path of length in the dual graph of formed by its
faces. This bound is tight up to lower order terms. For the bicolored version,
we describe an example of a line arrangement with blue and red lines
with no alternating path longer than . Further, we show that any line
arrangement with lines has a coloring such that it has an alternating path
of length . Our results also hold for pseudoline
arrangements.Comment: 19 page
Coloring problems on arrangements of pseudolines
Arrangements of pseudolines are a widely studied generalization of line
arrangements. They are defined as a finite family of infinite curves in the
Euclidean plane, any two of which intersect at exactly one point. One can state
various related coloring problems depending on the number of pseudolines.
In this article, we show that colors are sufficient for coloring the
crossings avoiding twice the same color on the boundary of any cell, or,
alternatively, avoiding twice the same color along any pseudoline. We also
study the problem of coloring the pseudolines avoiding monochromatic crossings.Comment: An extended abstract of this article was submitted to EuroCG202
Coloring and guarding arrangements
Given a simple arrangement of lines in the plane, what is the minimum number c of colors required so that we can color all lines in a way that no cell of the arrangement is monochromatic? In this paper we give worst-case bounds on the number c for both the above question and for some of its variations. Line coloring problems can be redefined as geometric hypergraph coloring problems as follows: if we define Hline-cell as the hypergraph whose vertices are lines and edges are cells of the arrangement, then c is equal to the chromatic number of this hypergraph. Specifically, we prove that this chromatic number is between Ω(log n= log log n) and O( √n). Furthermore, we give bounds on the minimum size of a subset S of the intersection points between pairs of lines in A such that every cell contains at least a vertex of S. This may be seen as the problem of 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 Hvertex-cell, 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 Hcell-zone hypergraph