2,129 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
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
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
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
How to Guard an Art Gallery: A Simple Mathematical Problem
The art gallery problem is a geometry question that seeks to find the minimum number of guards necessary to guard an art gallery based on the qualities of the museum’s shape, specifically the number of walls. Solved by Václav Chvátal in 1975, the resulting Art Gallery Theorem dictates that ⌊n/3⌋ guards are always sufficient and sometimes necessary to guard an art gallery with n walls. This theorem, along with the argument that proves it, are accessible and interesting results even to one with little to no mathematical knowledge, introducing readers to common concepts in both geometry and graph theory. Furthermore, the Art Gallery Theorem and its proof have many extensions, leading to other related theorems on guarding galleries, as well as various applications, including the use of its methods in robotics and GPS. This paper serves as a cursory introduction to the theorem, its most commonly referenced proof, and these extensions and applications, particularly for those with little familiarity with visibility problems or mathematics in general
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