## Abstract

Given an integer k≥2, we consider the problem of computing the smallest real number t(k) such that for each set P of points in the plane, there exists a t(k)-spanner for P that has chromatic number at most k. We prove that t(2)=3, t(3)=2, t(4)=2, and give upper and lower bounds on t(k) for k>4. We also show that for any >0, there exists a (1+)t(k)-spanner for P that has O(|P|) edges and chromatic number at most k. Finally, we consider an on-line variant of the problem where the points of P are given one after another, and the color of a point must be assigned at the moment the point is given. In this setting, we prove that t(2)=3, t(3)=1+3, t(4)=1+2, and give upper and lower bounds on t(k) for k>4.

Original language | English |
---|---|

Pages (from-to) | 134-146 |

Number of pages | 13 |

Journal | Computational Geometry: Theory and Applications |

Volume | 42 |

Issue number | 2 |

DOIs | |

State | Published - 1 Feb 2009 |

Externally published | Yes |

## Keywords

- Computational geometry
- Geometric graph
- Online algorithm
- Paz graph
- Spanners
- k-colorable graphs

## ASJC Scopus subject areas

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