Journal of Mathematical Sciences

, Volume 236, Issue 5, pp 554–578 | Cite as

Turán-Type Results for Distance Graphs in an Infinitesimal Plane Layer

  • L. E. ShabanovEmail author

In this paper, we obtain a lower bound on the number of edges in a unit distance graph Γ in an infinitesimal plane layer 2 × [0, ε]d, which relates the number of edges e(Γ), the number of vertices ν(Γ), and the independence number α(Γ). Our bound \( e\left(\varGamma \right)\ge \frac{19\nu \left(\varGamma \right)-50\alpha \left(\varGamma \right)}{3} \) is a generalization of a previous bound for distance graphs in the plane and a strong improvement of Turán’s bound in the case where \( \frac{1}{5}\le \frac{\alpha \left(\varGamma \right)}{v\left(\varGamma \right)}\le \frac{2}{7} \).


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    P. Brass, W. Moser, and J. Pach, Research Problems in Discrete Geometry, Springer (2005).Google Scholar
  2. 2.
    A. Dainyak and A. Sapozhenko, “Independent sets in graphs,” Discrete Math. Appl., 26, 323–346 (2016).MathSciNetCrossRefGoogle Scholar
  3. 3.
    P. Erdős, “On sets of distances of n points,” Amer. Math. Monthly, 53, 248–250 (1946).MathSciNetCrossRefGoogle Scholar
  4. 4.
    A. M. Raigorodskii, “Cliques and cycles in distance graphs and graphs of diameters,” in: Discrete Geometry and Algebraic Combinatorics, Contemp. Math., 625, Amer. Math. Soc., Providence, Rhode Island (2014), pp. 93–109.Google Scholar
  5. 5.
    A. M. Raigorodskii, “Coloring distance graphs and graphs of diameters,” in: Thirty Essays on Geometric Graph Theory, Springer, New York (2013), pp. 429–460.Google Scholar
  6. 6.
    A. M. Raigorodskii, “Combinatorial geometry and coding theory,” Fund. Inform., 145, 359–369 (2016).MathSciNetCrossRefGoogle Scholar
  7. 7.
    L. E. Shabanov and A. M. Raigorodskii, “Turán type results for distance graphs,” Discrete Comput. Geom., 56, 814–832 (2016).MathSciNetCrossRefGoogle Scholar
  8. 8.
    A. Soifer, Mathemetical Coloring Book, Springer (2009).Google Scholar
  9. 9.
    M. Tikhomirov, “On computational complexity of length embeddability of graphs,” Discrete Math., 339, No. 11, 2605–2612 (2016).MathSciNetCrossRefGoogle Scholar
  10. 10.
    P. Turán, “On an extremal problem in graph theory,” Mat. Fiz. Lapok, 48, 436–452 (1941).MathSciNetGoogle Scholar
  11. 11.
    A. E. Guterman, V. K. Lubimov, A. M. Raigorodskii, and A. S. Usachev, “On the independence numbers of distance graphs with vertices at {−1, 0, 1}n,” Mat. Zametki, 86, No. 5, 794–796 (2009).MathSciNetCrossRefGoogle Scholar
  12. 12.
    A. Y. Kanel-Belov, V. A. Voronov, and D. D. Cherkashin, “On the chromatic number of the plane,” Algebra Analiz, 29, No. 5 (2017).Google Scholar
  13. 13.
    V. K. Lubimov and A. M. Raigorodskii, “Lower bounds for the independence numbers of some distance graphs with vertices at {−1, 0, 1}n,” Dokl. Akad. Nauk, 427, No. 4, 458–460 (2009).MathSciNetGoogle Scholar
  14. 14.
    E. I. Ponomarenko and A. M. Raigorodskii, “New upper bounds for the independence numbers of graphs with vertices in {−1, 0, 1}n and their applications to problems of the chromatic numbers of distance graphs,” Mat. Zametki, 96, No. 1, 138–147 (2014).MathSciNetCrossRefGoogle Scholar
  15. 15.
    A. M. Raigorodskii, “The Erdős–Hadwiger problem and the chromatic numbers of finite geometric graphs,” Mat. Sb., 196, No. 1, 123–156 (2005).MathSciNetCrossRefGoogle Scholar
  16. 16.
    A. A. Sagdeev and A. M. Raigorodskii, “On the chromatic number of a space with a forbidden regular simplex,” Dokl. Acad. Nauk, 472, No. 2, 127–129 (2017).MathSciNetzbMATHGoogle Scholar
  17. 17.
    M. Tikhomirov, “On the problem of testing the distance and multidistance embeddability of a graph,” Dokl. Akad. Nauk, 468, No. 3, 261–263 (2016).MathSciNetGoogle Scholar
  18. 18.
    D. D. Cherkashin and A. M. Raigorodskii, “On the chromatic numbers of spaces of small dimension,” Dokl. Akad. Nauk, 472, No. 1, 11–12 (2017).MathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Moscow Institute of Physics and TechnologyDolgoprudnyRussia

Personalised recommendations