### Abstract

We prove that every finite set of homothetic copies of a given convex body in the plane can be colored with four colors so that any point covered by at least two copies is covered by two copies with distinct colors. This generalizes a previous result from Smorodinsky (SIAM J. Disc. Math. 2007). Then we show that for any k≥2, every three-dimensional hypergraph can be colored with 6(k-1) colors so that every hyperedge e contains min{|e|,k} vertices with mutually distinct colors. This refines a previous result from Aloupis et al. (Disc. & Comp. Geom. 2009). As corollaries, we improve on previous results for conflict-free coloring, k-strong conflict-free coloring, and choosability. Proofs of the upper bounds are constructive and yield simple, polynomial-time algorithms.

Original language | English |
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Pages (from-to) | 1027-1035 |

Number of pages | 9 |

Journal | Computational Geometry: Theory and Applications |

Volume | 46 |

Issue number | 9 |

DOIs | |

Publication status | Published - 2013 Jul 19 |

Externally published | Yes |

### Keywords

- 4-Color Theorem
- Chromatic numbers
- Conflict-free coloring
- Geometric hypergraphs
- Hypergraph coloring

### ASJC Scopus subject areas

- Computer Science Applications
- Geometry and Topology
- Control and Optimization
- Computational Theory and Mathematics
- Computational Mathematics

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## Cite this

*Computational Geometry: Theory and Applications*,

*46*(9), 1027-1035. https://doi.org/10.1016/j.comgeo.2013.06.004