24 research outputs found
Convex Polygons are Self-Coverable
We introduce a new notion for geometric families called self-coverability and
show that homothets of convex polygons are self-coverable. As a corollary, we
obtain several results about coloring point sets such that any member of the
family with many points contains all colors. This is dual (and in some cases
equivalent) to the much investigated cover-decomposability problem
More on Decomposing Coverings by Octants
In this note we improve our upper bound given earlier by showing that every
9-fold covering of a point set in the space by finitely many translates of an
octant decomposes into two coverings, and our lower bound by a construction for
a 4-fold covering that does not decompose into two coverings. The same bounds
also hold for coverings of points in by finitely many homothets or
translates of a triangle. We also prove that certain dynamic interval coloring
problems are equivalent to the above question
Making Octants Colorful and Related Covering Decomposition Problems
We give new positive results on the long-standing open problem of geometric
covering decomposition for homothetic polygons. In particular, we prove that
for any positive integer k, every finite set of points in R^3 can be colored
with k colors so that every translate of the negative octant containing at
least k^6 points contains at least one of each color. The best previously known
bound was doubly exponential in k. This yields, among other corollaries, the
first polynomial bound for the decomposability of multiple coverings by
homothetic triangles. We also investigate related decomposition problems
involving intervals appearing on a line. We prove that no algorithm can
dynamically maintain a decomposition of a multiple covering by intervals under
insertion of new intervals, even in a semi-online model, in which some coloring
decisions can be delayed. This implies that a wide range of sweeping plane
algorithms cannot guarantee any bound even for special cases of the octant
problem.Comment: version after revision process; minor changes in the expositio
Proper Coloring of Geometric Hypergraphs
We study whether for a given planar family F there is an m such that any finite set of points can be 3-colored such that any member of F that contains at least m points contains two points with different colors. We conjecture that if F is a family of pseudo-disks, then m=3 is sufficient. We prove that when F is the family of all homothetic copies of a given convex polygon, then such an m exists. We also study the problem in higher dimensions
Proper coloring of geometric hypergraphs
We study whether for a given planar family F there is an m such that any finite set of points can be 3-colored such that any member of F that contains at least m points contains two points with different colors. We conjecture that if F is a family of pseudo-disks, then m = 3 is sufficient. We prove that when F is the family of all homothetic copies of a given convex polygon, then such an m exists. We also study the problem in higher dimensions. © Balázs Keszegh and Dömötör Pálvölgyi
Hitting sets and colorings of hypergraphs
In this paper we study the minimal size of edges in hypergraph families which
guarantees the existence of a polychromatic coloring, that is, a -coloring
of a vertex set such that every hyperedge contains a vertex of all color
classes. We also investigate the connection of this problem with -shallow
hitting sets: sets of vertices that intersect each hyperedge in at least one
and at most vertices.
We determine in some hypergraph families the minimal for which a
-shallow hitting set exists.
We also study this problem for a special hypergraph family, which is induced
by arithmetic progressions with a difference from a given set. We show
connections between some geometric hypergraph families and the latter, and
prove relations between the set of differences and polychromatic colorability
Coloring Intersection Hypergraphs of Pseudo-Disks
We prove that the intersection hypergraph of a family of n pseudo-disks with respect to another family of pseudo-disks admits a proper coloring with 4 colors and a conflict-free coloring with O(log n) colors. Along the way we prove that the respective Delaunay-graph is planar. We also prove that the intersection hypergraph of a family of n regions with linear union complexity with respect to a family of pseudo-disks admits a proper coloring with constantly many colors and a conflict-free coloring with O(log n) colors. Our results serve as a common generalization and strengthening of many earlier results, including ones about proper and conflict-free coloring points with respect to pseudo-disks, coloring regions of linear union complexity with respect to points and coloring disks with respect to disks
Proper Coloring of Geometric Hypergraphs
We study whether for a given planar family F there is an m such that any finite set of points can be 3-colored so that anymember ofF that contains at leastm points contains two points with different colors. We conjecture that if F is a family of pseudo-disks, then such an m exists. We prove this in the special case when F is the family of all homothetic copies of a given convex polygon. We also study the problem in higher dimensions