Coloring intersection hypergraphs of pseudo-disks

Balázs Keszegh

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arXiv · 1711.05473

Coloring intersection hypergraphs of pseudo-disks

Balázs Keszegh

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Abstract

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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Balázs Keszegh. 2017-11-15. Coloring intersection hypergraphs of pseudo-disks. https://arxiv.org/abs/1711.05473

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