Graphs and Combinatorics

, Volume 4, Issue 1, pp 207–217

Repeated distances in space

  • David Avis
  • Paul Erdös
  • János Pach
Article

Abstract

Fori = 1,...,n letC(xi, ri) be a circle in the plane with centrexi and radiusri. A repeated distance graph is a directed graph whose vertices are the centres and where (xi, xj) is a directed edge wheneverxj lies on the circle with centrexi. Special cases are the nearest neighbour graph, whenri is the minimum distance betweenxi 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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References

  1. 1.
    Avis, D.: The number of furthest neighbour Pairs of a finite planar set. Amer. Math. Mon.91, 417–420 (1984)Google Scholar
  2. 2.
    Beck, J.: On the Lattice problem in the plane and some problems of Dirac, Motzkin, and Erdös in combinatorial geometry. Combinatorica3, 281–297 (1983)Google Scholar
  3. 3.
    Bollobás, B.: Extremal Graph Theory. London: Academic Press 1978Google Scholar
  4. 4.
    Chung, F.R.K., Szemerédi, E., Trotter W.T., Jr: Unpublished manuscript 1984Google Scholar
  5. 5.
    Erdös, P.: On sets of distances onn points in Euclidean space. Magyar Tud. Akad. Mat. Kutató Int. Kozl5, 165–169 (1960)Google Scholar
  6. 6.
    Grünbaum, B.: A proof of Vázsonyi's conjecture. Bull. Res. Council Israel Sect. A6, 77–78 (1956)Google Scholar
  7. 7.
    Harborth, H.: Solution to problem 664a. Elem. Math.29, 14–15 (1974)Google Scholar
  8. 8.
    Heppes, A.: Beweis einer Vermütung von A. Vázsonyi. Acta Math. Acad. Sci. Hung.7, 463–466 (1956)Google Scholar
  9. 9.
    Kövari, P., Sós, V.T., Túran, P.: On a problem of K. Zarankiewicz.3, 50–57 (1954)Google Scholar
  10. 10.
    Moser, W., Pach, J.: Research Problems in Discrete Geometry. Department of Mathematics, McGill University 1985Google Scholar
  11. 11.
    Straszewicz, S.: Bull. Acad. Pol. Sci. Cl. III5, 33–34 (1957)Google Scholar
  12. 12.
    Sutherland, J.W.: Solution to problem 167. Jahresber. Deutsch. Math. Verein.45, 33–34 (1935)Google Scholar
  13. 13.
    Toussaint, G.T.: Pattern recognition and geometric complexity. In: Proceedings 5th International Conference on Pattern Recognition, pp. 1–24. Miami Beach 1980Google Scholar

Copyright information

© Springer-Verlag 1988

Authors and Affiliations

  • David Avis
    • 1
  • Paul Erdös
    • 2
  • János Pach
    • 2
  1. 1.School of Computer ScienceMcGill UniversityMontrealCanada
  2. 2.Mathematical Institute of the Hungarian Academy of SciencesBudapestHungary

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