### 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(4ln 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. Lower bounds are also given for all of the above problems. 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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Title of host publication | LATIN 2010 |

Subtitle of host publication | Theoretical Informatics - 9th Latin American Symposium, Proceedings |

Pages | 2-13 |

Number of pages | 12 |

DOIs | |

Publication status | Published - 2010 Jun 18 |

Event | 9th Latin American Theoretical Informatics Symposium, LATIN 2010 - Oaxaca, Mexico Duration: 2010 Apr 19 → 2010 Apr 23 |

### Publication series

Name | Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) |
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Volume | 6034 LNCS |

ISSN (Print) | 0302-9743 |

ISSN (Electronic) | 1611-3349 |

### Other

Other | 9th Latin American Theoretical Informatics Symposium, LATIN 2010 |
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Country | Mexico |

City | Oaxaca |

Period | 10/4/19 → 10/4/23 |

### ASJC Scopus subject areas

- Theoretical Computer Science
- Computer Science(all)

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## Cite this

*LATIN 2010: Theoretical Informatics - 9th Latin American Symposium, Proceedings*(pp. 2-13). (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 6034 LNCS). https://doi.org/10.1007/978-3-642-12200-2_2