Repeated distances in space

David Avis, P. Erdős, János Pach

Research output: Contribution to journalArticle

11 Citations (Scopus)

Abstract

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.

Original languageEnglish
Pages (from-to)207-217
Number of pages11
JournalGraphs and Combinatorics
Volume4
Issue number1
DOIs
Publication statusPublished - Dec 1988

Fingerprint

Graph theory
Directed graphs
Distance Graph
Circle
Distance in Graphs
Extremal Graph Theory
Nearest Neighbor Graph
Hypersphere
Minimum Distance
Graph in graph theory
Directed Graph
Three-dimension
Radius
Generalise

ASJC Scopus subject areas

  • Discrete Mathematics and Combinatorics
  • Mathematics(all)

Cite this

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

In: Graphs and Combinatorics, Vol. 4, No. 1, 12.1988, p. 207-217.

Research output: Contribution to journalArticle

Avis, David ; Erdős, P. ; Pach, János. / Repeated distances in space. In: Graphs and Combinatorics. 1988 ; Vol. 4, No. 1. pp. 207-217.
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