### Abstract

For a set S of points in the plane, let d_{1}>d_{2}>... denote the different distances determined by S. Consider the graph G(S, k) whose vertices are the elements of S, and two are joined by an edge iff their distance is at least d_{k}. It is proved that the chromatic number of G(S, k) is at most 7 if |S|≥const k^{2}. If S consists of the vertices of a convex polygon and |S|≥const k^{2}, then the chromatic number of G(S, k) is at most 3. Both bounds are best possible. If S consists of the vertices of a convex polygon then G(S, k) has a vertex of degree at most 3 k - 1. This implies that in this case the chromatic number of G(S, k) is at most 3 k. The best bound here is probably 2 k+1, which is tight for the regular (2 k+1)-gon.

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

Pages (from-to) | 541-549 |

Number of pages | 9 |

Journal | Discrete & Computational Geometry |

Volume | 4 |

Issue number | 1 |

DOIs | |

Publication status | Published - Dec 1989 |

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### ASJC Scopus subject areas

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

### Cite this

*Discrete & Computational Geometry*,

*4*(1), 541-549. https://doi.org/10.1007/BF02187746

**On the graph of large distances.** / Erdős, P.; Lovász, L.; Vesztergombi, K.

Research output: Contribution to journal › Article

*Discrete & Computational Geometry*, vol. 4, no. 1, pp. 541-549. https://doi.org/10.1007/BF02187746

}

TY - JOUR

T1 - On the graph of large distances

AU - Erdős, P.

AU - Lovász, L.

AU - Vesztergombi, K.

PY - 1989/12

Y1 - 1989/12

N2 - For a set S of points in the plane, let d1>d2>... denote the different distances determined by S. Consider the graph G(S, k) whose vertices are the elements of S, and two are joined by an edge iff their distance is at least dk. It is proved that the chromatic number of G(S, k) is at most 7 if |S|≥const k2. If S consists of the vertices of a convex polygon and |S|≥const k2, then the chromatic number of G(S, k) is at most 3. Both bounds are best possible. If S consists of the vertices of a convex polygon then G(S, k) has a vertex of degree at most 3 k - 1. This implies that in this case the chromatic number of G(S, k) is at most 3 k. The best bound here is probably 2 k+1, which is tight for the regular (2 k+1)-gon.

AB - For a set S of points in the plane, let d1>d2>... denote the different distances determined by S. Consider the graph G(S, k) whose vertices are the elements of S, and two are joined by an edge iff their distance is at least dk. It is proved that the chromatic number of G(S, k) is at most 7 if |S|≥const k2. If S consists of the vertices of a convex polygon and |S|≥const k2, then the chromatic number of G(S, k) is at most 3. Both bounds are best possible. If S consists of the vertices of a convex polygon then G(S, k) has a vertex of degree at most 3 k - 1. This implies that in this case the chromatic number of G(S, k) is at most 3 k. The best bound here is probably 2 k+1, which is tight for the regular (2 k+1)-gon.

UR - http://www.scopus.com/inward/record.url?scp=34249972313&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=34249972313&partnerID=8YFLogxK

U2 - 10.1007/BF02187746

DO - 10.1007/BF02187746

M3 - Article

AN - SCOPUS:34249972313

VL - 4

SP - 541

EP - 549

JO - Discrete and Computational Geometry

JF - Discrete and Computational Geometry

SN - 0179-5376

IS - 1

ER -