### Abstract

We study the following geometric hypergraph coloring problem: given a planar point set and an integer k, we wish to color the points with k colors so that any axis-aligned strip containing sufficiently many points contains all colors. We show that if the strip contains at least 2k-1 points, such a coloring can always be found. In dimension d, we show that the same holds provided the strip contains at least k(4 ln k + ln d) points. We also consider the dual problem of coloring a given set of axis-aligned strips so that any sufficiently covered point in the plane is covered by k colors. We show that in dimension d the required coverage is at most d(k-1) + 1. This complements recent impossibility results on decomposition of strip coverings with arbitrary orientations. From the computational point of view, we show that deciding whether a three-dimensional point set can be 2-colored so that any strip containing at least three points contains both colors is NP-complete. This shows a big contrast with the planar case, for which this decision problem is easy.

Original language | English |
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Pages (from-to) | 327-339 |

Number of pages | 13 |

Journal | Graphs and Combinatorics |

Volume | 27 |

Issue number | 3 |

DOIs | |

Publication status | Published - 2011 May 1 |

Externally published | Yes |

### Keywords

- Computational geometry
- Covering decomposition
- Hypergraph coloring
- Lovász local lemma

### ASJC Scopus subject areas

- Theoretical Computer Science
- Discrete Mathematics and Combinatorics

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## Cite this

*Graphs and Combinatorics*,

*27*(3), 327-339. https://doi.org/10.1007/s00373-011-1014-5