Graphs and Combinatorics

, Volume 12, Issue 3, pp 295–303 | Cite as

On the connectivity of unit distance graphs

  • Michael Reid
Article

Abstract

For a number field K ⊆ ℝ, consider the graph G(Kd), whose vertices are elements of Kd, with an edge between any two points at (Euclidean) distance 1. We show that G(K2) is not connected whileG(Kd) is connected ford ≥ 5. We also give necessary and sufficient conditions for the connectedness of G(K3) and G(K4).

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Atiyah, M., MacDonald, G.: Introduction to Commutative Algebra. Addison-Wesley, Reading (1969)MATHGoogle Scholar
  2. 2.
    Benda, M., Perles, M.: Colorings of metric spaces (unpublished)Google Scholar
  3. 3.
    de Bruijn, N.G., Erdös, P.: A colour problem for infinite graphs and a problem in the theory of relations. Nederl. Akad. Wetensch. Indag. Math.13, 371–373 (1951)Google Scholar
  4. 4.
    Erdös, P.: Some new problems and results in graph theory and other branches of combinatorial mathematics. In: Combinatorics and Graph Theory, Lect. Notes in Math., Vol. 885, New York: Springer-VerlagGoogle Scholar
  5. 5.
    Fischer, K.: Additivek-colorable extensions of the rational plane. Discrete Math.82, 181–195 (1990)CrossRefMATHMathSciNetGoogle Scholar
  6. 6.
    Fischer, K.: The connected components of the graph ℚ(√N 1,...,√N d). Congr. Numer.72 213–221 (1990)MathSciNetGoogle Scholar
  7. 7.
    Jacobson, N.: Basic Algebra I, Second Edition. New York: W.H. Freeman and Co. 1985MATHGoogle Scholar
  8. 8.
    Johnson, P.D. Jr.: Two-colorings of real quadratic extensions of ℚ2 that forbid many distances. Congr. Numer.60, 51–58 (1987)MathSciNetGoogle Scholar
  9. 9.
    Lam, T.Y.: The Algebraic Theory of Quadratic Forms. Reading, Benjamin: 1973MATHGoogle Scholar
  10. 10.
    Marcus, D.A.: Number Fields. Universitext, New York: Springer-Verlag 1977CrossRefGoogle Scholar
  11. 11.
    Woodall, D.R.: Distances realized by sets covering the plane. J. Combin. Theory Ser.A 14, 187–200 (1973)CrossRefMATHMathSciNetGoogle Scholar
  12. 12.
    Zaks, J.: On the connectedness of some geometric graphs. J. Combin. Theory Ser.B 49, 143–150 (1990)CrossRefMATHMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag 1996

Authors and Affiliations

  • Michael Reid
    • 1
  1. 1.Department of MathematicsUniversity of CaliforniaBerkeleyUSA
  2. 2.SomervilleUSA

Personalised recommendations