365 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
The Szemeredi-Trotter Theorem in the Complex Plane
It is shown that points and lines in the complex Euclidean plane
determine point-line incidences. This
bound is the best possible, and it generalizes the celebrated theorem by
Szemer\'edi and Trotter about point-line incidences in the real Euclidean plane
.Comment: 24 pages, 5 figures, to appear in Combinatoric
Decomposition of Multiple Coverings into More Parts
We prove that for every centrally symmetric convex polygon Q, there exists a
constant alpha such that any alpha*k-fold covering of the plane by translates
of Q can be decomposed into k coverings. This improves on a quadratic upper
bound proved by Pach and Toth (SoCG'07). The question is motivated by a sensor
network problem, in which a region has to be monitored by sensors with limited
battery lifetime
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
Conflict-free connection numbers of line graphs
A path in an edge-colored graph is called \emph{conflict-free} if it contains
at least one color used on exactly one of its edges. An edge-colored graph
is \emph{conflict-free connected} if for any two distinct vertices of ,
there is a conflict-free path connecting them. For a connected graph , the
\emph{conflict-free connection number} of , denoted by , is defined
as the minimum number of colors that are required to make conflict-free
connected. In this paper, we investigate the conflict-free connection numbers
of connected claw-free graphs, especially line graphs. We first show that for
an arbitrary connected graph , there exists a positive integer such that
. Secondly, we get the exact value of the conflict-free
connection number of a connected claw-free graph, especially a connected line
graph. Thirdly, we prove that for an arbitrary connected graph and an
arbitrary positive integer , we always have , with only the exception that is isomorphic to a star of order
at least~ and . Finally, we obtain the exact values of ,
and use them as an efficient tool to get the smallest nonnegative integer
such that .Comment: 11 page
Overlap properties of geometric expanders
The {\em overlap number} of a finite -uniform hypergraph is
defined as the largest constant such that no matter how we map
the vertices of into , there is a point covered by at least a
-fraction of the simplices induced by the images of its hyperedges.
In~\cite{Gro2}, motivated by the search for an analogue of the notion of graph
expansion for higher dimensional simplicial complexes, it was asked whether or
not there exists a sequence of arbitrarily large
-uniform hypergraphs with bounded degree, for which . Using both random methods and explicit constructions, we answer this
question positively by constructing infinite families of -uniform
hypergraphs with bounded degree such that their overlap numbers are bounded
from below by a positive constant . We also show that, for every ,
the best value of the constant that can be achieved by such a
construction is asymptotically equal to the limit of the overlap numbers of the
complete -uniform hypergraphs with vertices, as
. For the proof of the latter statement, we establish the
following geometric partitioning result of independent interest. For any
and any , there exists satisfying the
following condition. For any , for any point and
for any finite Borel measure on with respect to which
every hyperplane has measure , there is a partition into measurable parts of equal measure such that all but
at most an -fraction of the -tuples
have the property that either all simplices with
one vertex in each contain or none of these simplices contain
On the Recognition of Fan-Planar and Maximal Outer-Fan-Planar Graphs
Fan-planar graphs were recently introduced as a generalization of 1-planar
graphs. A graph is fan-planar if it can be embedded in the plane, such that
each edge that is crossed more than once, is crossed by a bundle of two or more
edges incident to a common vertex. A graph is outer-fan-planar if it has a
fan-planar embedding in which every vertex is on the outer face. If, in
addition, the insertion of an edge destroys its outer-fan-planarity, then it is
maximal outer-fan-planar. In this paper, we present a polynomial-time algorithm
to test whether a given graph is maximal outer-fan-planar. The algorithm can
also be employed to produce an outer-fan-planar embedding, if one exists. On
the negative side, we show that testing fan-planarity of a graph is NP-hard,
for the case where the rotation system (i.e., the cyclic order of the edges
around each vertex) is given
Conflict-free colorings of graphs and hypergraphs Probability and Computing
A colouring of the vertices of a hypergraph H is called conflict-free if each hyperedge E of H contains a vertex of 'unique' colour that does not get repeated in E. The smallest number of colours required for such a colouring is called the conflict-free chromatic number of H, and is denoted by chi(CF)(H). This parameter wits first introduced by Even, Lotker, Ron and Smorodinsky (FOCS 2002) in a geometric setting, in connection with frequency assignment problems for cellular networks. Here we analyse this notion for general hypergraphs. It is shown that chi(CF)(H) = 3. Using Lovasz's Local Lemma, the same result holds for hypergraphs in which the size of every edge is at least 2t - 1 and every edge intersects at most tit others. We give efficient polynomial-time algorithms to obtain such colourings
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