On The Chromatic Number of Geometric Hypergraphs

Smorodinsky, Shakhar

doi:10.1137/050642368

Cited by 40 publications

(15 citation statements)

References 14 publications

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“…Proof of Corollary 17. As mentioned already, the proof uses the by now standard framework of [27]. Given the family B containing n regions, the first step is to color it properly wrt.…”

Section: Consequencesmentioning

confidence: 99%

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Coloring Intersection Hypergraphs of Pseudo-Disks

Keszegh

2019

Discrete Comput Geom

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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.

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“…Proof of Corollary 17. As mentioned already, the proof uses the by now standard framework of [27]. Given the family B containing n regions, the first step is to color it properly wrt.…”

Section: Consequencesmentioning

confidence: 99%

“…We also define the following subgraph of the Delaunay-graph: In the literature hypergraphs that have the property assumed in Observation 5 are sometimes called rank two hypergraphs (e.g., in [27]). Definition 6.…”

Section: Introductionmentioning

confidence: 99%

Coloring Intersection Hypergraphs of Pseudo-Disks

Keszegh

2019

Discrete Comput Geom

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“…Clearly, colorful 2-colorings and proper 2-colorings coincide, but also for a larger number of colors proper colorings of geometric hypergraphs have been considered in the primal and dual setting. There are results for bottomless rectangles [17], halfplanes [15,17], octants [10], rectangles [12,1,26], and disks [28,33].…”

Section: Theorem 3 ([19]mentioning

confidence: 99%

Making Octants Colorful and Related Covering Decomposition Problems

Cardinal¹,

Knauer²,

Micek³

et al. 2014

SIAM J. Discrete Math.

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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.A preliminary version of this paper was presented at the 2014 ACM-SIAM Symposium on Discrete Algorithms (SODA'14).Journal version of this paper is submitted to SIAM J. Discrete Math.

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“…There are results for bottomless rectangles [17], halfplanes [15,17], octants [10], rectangles [12,1,26], and disks [28,33].…”

Section: Theorem 3 ([19]mentioning

confidence: 99%

Making Octants Colorful and Related Covering Decomposition Problems

Cardinal¹,

Knauer²,

Micek³

et al. 2013

Proceedings of the Twenty-Fifth Annual ACM-SIAM Symposium on Discrete Algorithms

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Abstract. 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.

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On The Chromatic Number of Geometric Hypergraphs

Cited by 40 publications

References 14 publications

Coloring Intersection Hypergraphs of Pseudo-Disks

Coloring Intersection Hypergraphs of Pseudo-Disks

Making Octants Colorful and Related Covering Decomposition Problems

Making Octants Colorful and Related Covering Decomposition Problems

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