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Repeated distances in space

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Abstract

Fori = 1,...,n letC(xi, ri) be a circle in the plane with centrex i and radiusr i. A repeated distance graph is a directed graph whose vertices are the centres and where (x i, xj) is a directed edge wheneverx j lies on the circle with centrex i. Special cases are the nearest neighbour graph, whenr i is the minimum distance betweenx 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 ind dimensions, with particularly tight bounds for the furthest neighbour graph in three dimensions. The proofs use extremal graph theory.

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Authors and Affiliations

  1. School of Computer Science, McGill University, H3A 2K6, Montreal, Canada

    David Avis

  2. Mathematical Institute of the Hungarian Academy of Sciences, 1364, Budapest, Hungary

    Paul Erdös & János Pach

Authors
  1. David Avis
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  2. Paul Erdös
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  3. János Pach
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Additional information

Research supported by the Natural Science and Engineering Research Council grant number A3013 and the F.C.A.R. grant number EQ1678.

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Avis, D., Erdös, P. & Pach, J. Repeated distances in space. Graphs and Combinatorics 4, 207–217 (1988). https://doi.org/10.1007/BF01864161

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  • DOI: https://doi.org/10.1007/BF01864161

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