16 research outputs found
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
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 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
Note on polychromatic coloring of hereditary hypergraph families
We exhibit a 5-uniform hypergraph that has no polychromatic 3-coloring, but
all its restricted subhypergraphs with edges of size at least 3 are
2-colorable. This disproves a bold conjecture of Keszegh and the author, and
can be considered as the first step to understand polychromatic colorings of
hereditary hypergraph families better since the seminal work of Berge. We also
show that our method cannot give hypergraphs of arbitrary high uniformity, and
mention some connections to panchromatic colorings
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