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Coloring Points with Respect to Squares

Authors Eyal Ackerman, Balázs Keszegh, Máté Vizer

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Eyal Ackerman
Balázs Keszegh
Máté Vizer

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Eyal Ackerman, Balázs Keszegh, and Máté Vizer. Coloring Points with Respect to Squares. In 32nd International Symposium on Computational Geometry (SoCG 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 51, pp. 5:1-5:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016) https://doi.org/10.4230/LIPIcs.SoCG.2016.5

Abstract

We consider the problem of 2-coloring geometric hypergraphs. Specifically, we show that there is a constant m such that any finite set S of points in the plane can be 2-colored such that every axis-parallel square that contains at least m points from S contains points of both colors. Our proof is constructive, that is, it provides a polynomial-time algorithm for obtaining such a 2-coloring. By affine transformations this result immediately applies also when considering homothets of a fixed parallelogram.

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Keywords
  • Geometric hypergraph coloring
  • Polychromatic coloring
  • Homothets
  • Cover-decomposability

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