22 research outputs found
Coloring Hypergraphs Induced by Dynamic Point Sets and Bottomless Rectangles
We consider a coloring problem on dynamic, one-dimensional point sets: points
appearing and disappearing on a line at given times. We wish to color them with
k colors so that at any time, any sequence of p(k) consecutive points, for some
function p, contains at least one point of each color.
We prove that no such function p(k) exists in general. However, in the
restricted case in which points appear gradually, but never disappear, we give
a coloring algorithm guaranteeing the property at any time with p(k)=3k-2. This
can be interpreted as coloring point sets in R^2 with k colors such that any
bottomless rectangle containing at least 3k-2 points contains at least one
point of each color. Here a bottomless rectangle is an axis-aligned rectangle
whose bottom edge is below the lowest point of the set. For this problem, we
also prove a lower bound p(k)>ck, where c>1.67. Hence for every k there exists
a point set, every k-coloring of which is such that there exists a bottomless
rectangle containing ck points and missing at least one of the k colors.
Chen et al. (2009) proved that no such function exists in the case of
general axis-aligned rectangles. Our result also complements recent results
from Keszegh and Palvolgyi on cover-decomposability of octants (2011, 2012).Comment: A preliminary version was presented by a subset of the authors to the
European Workshop on Computational Geometry, held in Assisi (Italy) on March
19-21, 201
Polychromatic Coloring for Half-Planes
We prove that for every integer , every finite set of points in the plane
can be -colored so that every half-plane that contains at least
points, also contains at least one point from every color class. We also show
that the bound is best possible. This improves the best previously known
lower and upper bounds of and respectively. We also show
that every finite set of half-planes can be colored so that if a point
belongs to a subset of at least of the half-planes then
contains a half-plane from every color class. This improves the best previously
known upper bound of . Another corollary of our first result is a new
proof of the existence of small size \eps-nets for points in the plane with
respect to half-planes.Comment: 11 pages, 5 figure
Indecomposable Coverings with Concave Polygons
We show that for any concave polygon that has no parallel sides and for any k, there is a k-fold covering of some point set by the translates of this polygon that cannot be decomposed into two coverings. Moreover, we give a complete classification of open polygons with this property. We also construct for any polytope (having dimension at least three) and for any k, a k-fold covering of the space by its translates that cannot be decomposed into two coverings
Advantage in the discrete Voronoi game
We study the discrete Voronoi game, where two players alternately claim vertices of a graph for t rounds. In the end, the remaining vertices are divided such that each player receives the vertices that are closer to his or her claimed vertices. We prove that there are graphs for which the second player gets almost all vertices in this game, but this is not possible for bounded-degree graphs. For trees, the first player can get at least one quarter of the vertices, and we give examples where she can get only little more than one third of them. We make some general observations, relating the result with many rounds to the result for the one-round game on the same graph
Unsplittable coverings in the plane
A system of sets forms an {\em -fold covering} of a set if every point
of belongs to at least of its members. A -fold covering is called a
{\em covering}. The problem of splitting multiple coverings into several
coverings was motivated by classical density estimates for {\em sphere
packings} as well as by the {\em planar sensor cover problem}. It has been the
prevailing conjecture for 35 years (settled in many special cases) that for
every plane convex body , there exists a constant such that every
-fold covering of the plane with translates of splits into
coverings. In the present paper, it is proved that this conjecture is false for
the unit disk. The proof can be generalized to construct, for every , an
unsplittable -fold covering of the plane with translates of any open convex
body which has a smooth boundary with everywhere {\em positive curvature}.
Somewhat surprisingly, {\em unbounded} open convex sets do not misbehave,
they satisfy the conjecture: every -fold covering of any region of the plane
by translates of such a set splits into two coverings. To establish this
result, we prove a general coloring theorem for hypergraphs of a special type:
{\em shift-chains}. We also show that there is a constant such that, for
any positive integer , every -fold covering of a region with unit disks
splits into two coverings, provided that every point is covered by {\em at
most} sets
An abstract approach to polychromatic coloring
The goal of this paper is to give a new, abstract approach to cover-decomposition and polychromatic colorings using hypergraphs on ordered vertex sets. We introduce an abstract version of a framework by Smorodinsky and Yuditsky, used for polychromatic coloring halfplanes, and apply it to so-called ABA-free hypergraphs, which are a generalization of interval graphs. Using our methods, we prove that (2k−1)-uniform ABA-free hypergraphs have a polychromatic k-coloring, a problem posed by the second author. We also prove the same for hypergraphs defined on a point set by pseudohalfplanes. These results are best possible. We also introduce several new notions that seem to be important for investigating polychromatic colorings and ϵ -nets, such as shallow hitting sets. We pose several open problems related to them. For example, is it true that given a finite point set S on a sphere and a set of halfspheres F, such that {S ∩ F | F ∈ F} is a Sperner family, we can select an R ⊂ S such that 1 ≤ |F ∩ R| ≤ 2 holds for every F ∈ F?. © Springer International Publishing Switzerland 2016
Advantage in the discrete Voronoi game
We study the discrete Voronoi game, where two players alternately claim vertices of a graph for t rounds. In the end, the remaining vertices are divided such that each player receives the vertices that are closer to his or her claimed vertices. We prove that there are graphs for which the second player gets almost all vertices in this game, but this is not possible for bounded-degree graphs. For trees, the first player can get at least one quarter of the vertices, and we give examples where she can get only little more than one third of them. We make some general observations, relating the result with many rounds to the result for the one-round game on the same graph
Convex polygons are cover-decomposable
We show that for any open convex polygon P, there is a constant k(P) such that any k(P)-fold covering of the plane with translates of P can be decomposed into two coverings