Skip to main content
SpringerLink
Log in

Diameters of Random Distance Graphs

  • Published:
Journal of Mathematical Sciences Aims and scope Submit manuscript

Abstract

This paper contains an almost exhaustive description of all cases where a distance random graph asymptotically almost surely has diameter 1, 2, or greater than 2.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. P. K. Agarwal and J. Pach, Combinatorial Geometry, Wiley, New York (1995).

    MATH  Google Scholar 

  2. N. Alon and J. H. Spencer, The Probabilistic Mathod, Wiley, New York etc. (2000).

    Book  Google Scholar 

  3. J. Balogh, A. V. Kostochka, and A. M. Raigorodskii, “Coloring some finite sets in ℝ n,” Discuss. Math. Graph Theory, 33, No. 1, 25–31 (2013).

    Article  MATH  MathSciNet  Google Scholar 

  4. L. Bassalygo, G. Cohen, and G. Zémor, “Codes with forbidden distances,” Discr. Math., 213, 3–11 (2000).

    Article  MATH  MathSciNet  Google Scholar 

  5. L. I. Bogolyubsky, A. S. Gusev, M. M. Pyaderkin, and A. M. Raigorodskii, “Independence numbers and chromatic numbers of random subgraphs in some sequences of graphs,” Dokl. Ross. Akad. Nauk, 457, No. 4, 383–387 (2014).

    MATH  MathSciNet  Google Scholar 

  6. L. I. Bogolyubsky, A. S. Gusev, M. M. Pyaderkin, and A. M. Raigorodskii, “Independence numbers and chromatic numbers of random subgraphs of some distance graphs,” Mat. Sb., 206, No. 10, 3–36 (2015).

    Article  MATH  MathSciNet  Google Scholar 

  7. B. Bollobás, Random Graphs, Cambridge Univ. Press (2001).

  8. B. Bollobás, B. P. Narayanan, and A. M. Raigorodskii, “On the stability of the Erdős–Ko–Rado theorem,” J. Comb. Theory, Ser. A, 137, 64–78 (2016).

    Article  MATH  Google Scholar 

  9. V. G. Boltyanski, H. Martini, and P. S. Soltan, Excursions into Combinatorial Geometry, Universitext, Springer-Verlag, Berlin (1997).

    Book  MATH  Google Scholar 

  10. E. E. Demekhin, A. M. Raigorodskii, and O. I. Rubanov, “Distance graphs having large chromatic numbers and containing no cliques or cycles of a given size,” Mat. Sb., 204, No. 4, 49–78 (2013).

    Article  MATH  Google Scholar 

  11. R. L. Graham, B. L. Rothschild, and J. H. Spencer, Ramsey Theory, Wiley, New York (1990).

    MATH  Google Scholar 

  12. S. Janson, T. Łuczak, and A. Ruciński, Random Graphs, Wiley, New York (2000).

    Book  MATH  Google Scholar 

  13. V. Klee and S. Wagon, Old and New Unsolved Problems in Plane Geometry and Number Theory, Math. Association of America (1991).

  14. V. F. Kolchin, Random Graphs [in Russian], Fizmatlit, Moscow (2002).

    Google Scholar 

  15. M. Krivelevich and B. Sudakov, “Pseudo-random graphs,” in: More Sets, Graphs, and Numbers (E. Győri, G. O. H. Katona, and L. Lovász, eds.), Bolyai Soc. Math. Stud., 15 (2006), pp. 199–262.

  16. F. J. MacWilliams and N. J. A. Sloane, The Theory of Error-Correcting Codes, North-Holland, Amsterdam–New York–Oxford (1977).

  17. J. Matoušek, Using the Borsuk–Ulam Theorem, Universitext, Springer-Verlag, Berlin (2003).

  18. K. A. Mikhailov and A. M. Raigorodskii, “On the Ramsey numbers for complete distance graphs with vertices in {0, 1} n,” Mat. Sb., 200, No. 12, 63–80 (2009).

    Article  MATH  MathSciNet  Google Scholar 

  19. Z. Nagy, “A certain constructive estimate of the Ramsey number,” Mat. Lapok, 23, Nos. 301–302, 26 (1972).

    MathSciNet  Google Scholar 

  20. A. M. Raigorodskii, “Borsuk’s problem and the chromatic numbers of some metric spaces,” Usp. Mat. Nauk, 56, No. 1, 107–146 (2001).

    Article  MATH  MathSciNet  Google Scholar 

  21. A. M. Raigorodskii, Three Lectures on the Borsuk Partition Problem, London Math. Soc. Lect. Note Ser., 347, 202–248 (2007).

    MATH  MathSciNet  Google Scholar 

  22. A. M. Raigorodskii, “Around Borsuk’s Hypothesis,” J. Math. Sci., 154, No. 4, 604–623 (2008).

    Article  MATH  MathSciNet  Google Scholar 

  23. A. M. Raigorodskii, “On chromatic numbers of spheres in Euclidean spaces,” Dokl. Ross Akad. Nauk, 432, No. 2, 174–177 (2010).

    MATH  MathSciNet  Google Scholar 

  24. A. M. Raigorodskii, “On the chromatic numbers of spheres in ℝ n,” Combinatorica, 32, No. 1, 111–123 (2012).

    Article  MATH  MathSciNet  Google Scholar 

  25. A. M. Raigorodskii, “Coloring distance graphs and graphs of diameters,” in: Thirty Essays on Geometric Graph Theory (J. Pach, ed.), Springer-Verlag (2013), pp. 429–460.

  26. A. M. Raigorodskii, “Cliques and cycles in distance graphs and graphs of diameters,” Discr. Geom. Alg. Combin., 625, 93–109 (2014).

    MATH  MathSciNet  Google Scholar 

  27. A. Soifer, The Mathematical Coloring Book, Springer-Verlag (2009).

  28. L. A. Székely, “Erdős on unit distances and the Szemerédi–Trotter theorems”, in: Paul Erdős and His Mathematics, J. Bolyai Math. Soc., Springer-Verlag, 11 (2002), pp. 649–666.

  29. M. E. Zhukovskii and A. M. Raigorodskii, “Random graphs: models and asymptotic characteristics,” Usp. Mat. Nauk, 70, No. 1 (421), 35–88 (2015).

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Moscow Institute of Physics and Technology (State University), Moscow, Russia

    L. Iskhakov & M. Mironov

Authors
  1. L. Iskhakov
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. M. Mironov
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 100, Geometry and Mechanics, 2016.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Iskhakov, L., Mironov, M. Diameters of Random Distance Graphs. J Math Sci 227, 407–418 (2017). https://doi.org/10.1007/s10958-017-3594-y

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10958-017-3594-y

Search

Navigation