26 research outputs found
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 (2kâ1)-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 Ï” -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 â© F | F â F} is a Sperner family, we can select an R â S such that 1 †|F â© R| †2 holds for every F â F?. © Springer International Publishing Switzerland 2016
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
Planar Octilinear Drawings with One Bend Per Edge
In octilinear drawings of planar graphs, every edge is drawn as an
alternating sequence of horizontal, vertical and diagonal ()
line-segments. In this paper, we study octilinear drawings of low edge
complexity, i.e., with few bends per edge. A -planar graph is a planar graph
in which each vertex has degree less or equal to . In particular, we prove
that every 4-planar graph admits a planar octilinear drawing with at most one
bend per edge on an integer grid of size . For 5-planar
graphs, we prove that one bend per edge still suffices in order to construct
planar octilinear drawings, but in super-polynomial area. However, for 6-planar
graphs we give a class of graphs whose planar octilinear drawings require at
least two bends per edge
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
ADAPTIVE MAJORITY PROBLEMS FOR RESTRICTED QUERY GRAPHS AND FOR WEIGHTED SETS
Suppose that the vertices of a graph G are colored with two colors in an unknown way. The color that occurs on more than half of the vertices is called the majority color (if it exists), and any vertex of this color is called a majority vertex. We study the problem of finding a majority vertex (or show that none exists), if we can query edges to learn whether their endpoints have the same or different colors. Denote the least number of queries needed in the worst case by m(G). It was shown by Saks and Werman that m(K-n) = n - b(n) where b(n) is the number of 1's in the binary representation of n. In this paper we initiate the study of the problem for general graphs. The obvious bounds for a connected graph G on n vertices are n - b(n) <= m(G) <= n - 1. We show that for any tree T on an even number of vertices we have m(T) = n - 1, and that for any tree T on an odd number of vertices, we have n - 65 <= m (T) <= n - 2. Our proof uses results about the weighted version of the problem for K-n, which may be of independent interest. We also exhibit a sequence G(n) of graphs with m(G(n)) = n - b(n) such that the number of edges in G(n) is O(nb(n))
On Smooth Orthogonal and Octilinear Drawings: Relations, Complexity and Kandinsky Drawings
We study two variants of the well-known orthogonal drawing model: (i) the
smooth orthogonal, and (ii) the octilinear. Both models form an extension of
the orthogonal, by supporting one additional type of edge segments (circular
arcs and diagonal segments, respectively).
For planar graphs of max-degree 4, we analyze relationships between the graph
classes that can be drawn bendless in the two models and we also prove
NP-hardness for a restricted version of the bendless drawing problem for both
models. For planar graphs of higher degree, we present an algorithm that
produces bi-monotone smooth orthogonal drawings with at most two segments per
edge, which also guarantees a linear number of edges with exactly one segment.Comment: Appears in the Proceedings of the 25th International Symposium on
Graph Drawing and Network Visualization (GD 2017
Nonrepetitive Colouring via Entropy Compression
A vertex colouring of a graph is \emph{nonrepetitive} if there is no path
whose first half receives the same sequence of colours as the second half. A
graph is nonrepetitively -choosable if given lists of at least colours
at each vertex, there is a nonrepetitive colouring such that each vertex is
coloured from its own list. It is known that every graph with maximum degree
is -choosable, for some constant . We prove this result
with (ignoring lower order terms). We then prove that every subdivision
of a graph with sufficiently many division vertices per edge is nonrepetitively
5-choosable. The proofs of both these results are based on the Moser-Tardos
entropy-compression method, and a recent extension by Grytczuk, Kozik and Micek
for the nonrepetitive choosability of paths. Finally, we prove that every graph
with pathwidth is nonrepetitively -colourable.Comment: v4: Minor changes made following helpful comments by the referee