91 research outputs found
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
Conflict-Free Coloring of Intersection Graphs of Geometric Objects
In FOCS'2002, Even et al. introduced and studied the notion of conflict-free
colorings of geometrically defined hypergraphs. They motivated it by frequency
assignment problems in cellular networks. This notion has been extensively
studied since then.
A conflict-free coloring of a graph is a coloring of its vertices such that
the neighborhood (pointed or closed) of each vertex contains a vertex whose
color differs from the colors of all other vertices in that neighborhood. In
this paper we study conflict-colorings of intersection graphs of geometric
objects. We show that any intersection graph of n pseudo-discs in the plane
admits a conflict-free coloring with O(\log n) colors, with respect to both
closed and pointed neighborhoods. We also show that the latter bound is
asymptotically sharp. Using our methods, we also obtain a strengthening of the
two main results of Even et al. which we believe is of independent interest. In
particular, in view of the original motivation to study such colorings, this
strengthening suggests further applications to frequency assignment in wireless
networks.
Finally, we present bounds on the number of colors needed for conflict-free
colorings of other classes of intersection graphs, including intersection
graphs of axis-parallel rectangles and of \rho-fat objects in the plane.Comment: 18 page
Aligned plane drawings of the generalized Delaunay-graphs for pseudo-disks
We study general Delaunay-graphs, which are natural generalizations of
Delaunay triangulations to arbitrary families, in particular to pseudo-disks.
We prove that for any finite pseudo-disk family and point set, there is a plane
drawing of their Delaunay-graph such that every edge lies inside every
pseudo-disk that contains its endpoints
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
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
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