### Abstract

For i = 1,..., n let C(x_{i}, r_{i}) be a circle in the plane with centre x_{i} and radius r_{i}. A repeated distance graph is a directed graph whose vertices are the centres and where (x_{i}, x_{j}) is a directed edge whenever x_{j} lies on the circle with centre x_{i}. Special cases are the nearest neighbour graph, when r_{i} is the minimum distance between x_{i} and any other centre, and the furthest neighbour graph which is similar except that maximum replaces minimum. Repeated distance graphs generalize to any dimension with spheres or hyperspheres replacing circles. Bounds are given on the number of edges in repeated distance graphs in d dimensions, with particularly tight bounds for the furthest neighbour graph in three dimensions. The proofs use extremal graph theory.

Original language | English |
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Pages (from-to) | 207-217 |

Number of pages | 11 |

Journal | Graphs and Combinatorics |

Volume | 4 |

Issue number | 1 |

DOIs | |

Publication status | Published - Dec 1988 |

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### ASJC Scopus subject areas

- Discrete Mathematics and Combinatorics
- Mathematics(all)

### Cite this

*Graphs and Combinatorics*,

*4*(1), 207-217. https://doi.org/10.1007/BF01864161

**Repeated distances in space.** / Avis, David; Erdős, P.; Pach, János.

Research output: Contribution to journal › Article

*Graphs and Combinatorics*, vol. 4, no. 1, pp. 207-217. https://doi.org/10.1007/BF01864161

}

TY - JOUR

T1 - Repeated distances in space

AU - Avis, David

AU - Erdős, P.

AU - Pach, János

PY - 1988/12

Y1 - 1988/12

N2 - For i = 1,..., n let C(xi, ri) be a circle in the plane with centre xi and radius ri. A repeated distance graph is a directed graph whose vertices are the centres and where (xi, xj) is a directed edge whenever xj lies on the circle with centre xi. Special cases are the nearest neighbour graph, when ri is the minimum distance between xi and any other centre, and the furthest neighbour graph which is similar except that maximum replaces minimum. Repeated distance graphs generalize to any dimension with spheres or hyperspheres replacing circles. Bounds are given on the number of edges in repeated distance graphs in d dimensions, with particularly tight bounds for the furthest neighbour graph in three dimensions. The proofs use extremal graph theory.

AB - For i = 1,..., n let C(xi, ri) be a circle in the plane with centre xi and radius ri. A repeated distance graph is a directed graph whose vertices are the centres and where (xi, xj) is a directed edge whenever xj lies on the circle with centre xi. Special cases are the nearest neighbour graph, when ri is the minimum distance between xi and any other centre, and the furthest neighbour graph which is similar except that maximum replaces minimum. Repeated distance graphs generalize to any dimension with spheres or hyperspheres replacing circles. Bounds are given on the number of edges in repeated distance graphs in d dimensions, with particularly tight bounds for the furthest neighbour graph in three dimensions. The proofs use extremal graph theory.

UR - http://www.scopus.com/inward/record.url?scp=0043267913&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0043267913&partnerID=8YFLogxK

U2 - 10.1007/BF01864161

DO - 10.1007/BF01864161

M3 - Article

VL - 4

SP - 207

EP - 217

JO - Graphs and Combinatorics

JF - Graphs and Combinatorics

SN - 0911-0119

IS - 1

ER -