### Abstract

Continuing and extending the analysis in a previous paper [15], we establish several combinatorial results on the complexity of arrangements of circles in the plane. The main results are a collection of partial solutions to the conjecture that (a) any arrangement of unit circles with at least one intersecting pair has a vertex incident to at most three circles, and (b) any arrangement of circles of arbitrary radii with at least one intersecting pair has a vertex incident to at most three circles, under appropriate assumptions on the number of intersecting pairs of circles (see below for a more precise statement).

Original language | English (US) |
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Pages (from-to) | 465-492 |

Number of pages | 28 |

Journal | Discrete and Computational Geometry |

Volume | 26 |

Issue number | 4 |

DOIs | |

State | Published - Dec 2001 |

Externally published | Yes |

### All Science Journal Classification (ASJC) codes

- Theoretical Computer Science
- Geometry and Topology
- Discrete Mathematics and Combinatorics
- Computational Theory and Mathematics

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## Cite this

Alon, N., Last, H., Pinchasi, R., & Sharir, M. (2001). On the complexity of arrangements of circles in the plane.

*Discrete and Computational Geometry*,*26*(4), 465-492. https://doi.org/10.1007/s00454-001-0043-x