Skip to main content
SpringerLink
Log in

Hamiltonian paths in distance graphs

  • Published:
Mathematical Notes Aims and scope Submit manuscript

Abstract

The object of study is the graph

$G(n,r,s) = (V(n,r),E(n,r,s))$

with

$\begin{gathered} V(n,r) = \{ v:v \subset \{ 1,...,n\} ,|v| = r\} , \hfill \\ E(n,r,s) = \{ \{ v,u\} :v,u \in V(n,r),|v \cap u| = s\} ; \hfill \\ \end{gathered} $

i.e., the vertices of the graph are r-subsets of the set R n = {1, …, n}, and two vertices are connected by an edge if these vertices intersect in precisely s elements. Two-sided estimates for the number of Hamiltonian paths in the graph G(n, k, 1) as n → ∞ are obtained.

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. F. J. MacWilliams and N. J. A. Sloane, The Theory of Error Correcting Codes (North-Holland, Amsterdam-New York-Oxford, 1977; Radio i Svyaz’,Moscow, 1979).

    MATH  Google Scholar 

  2. P. Brass, W. Moser, and J. Pach, Research Problems in Discrete Geometry (Springer, New York, 2005).

    MATH  Google Scholar 

  3. A. M. Raigorodskii, “Borsuk’s problem and the chromatic numbers of some metric spaces,” Uspekhi Mat. Nauk 56(1), 107–146 (2001) [RussianMath. Surveys 56 (1), 103–139 (2001)].

    Article  MathSciNet  Google Scholar 

  4. A. M. Raigorodskii, “Coloring Distance Graphs and Graphs of Diameters,” in Thirty Essays on Geometric Graph Theory (Springer, New York, 2013), pp. 429–460.

    Chapter  Google Scholar 

  5. J. Pach and P. K. Agarwal, Combinatorial Geometry, in Wiley-Intersci. Ser. DiscreteMath. Optim. (John Wiley and Sons, New York, 1995).

    Google Scholar 

  6. A. M. Raigorodskii, “The problems of Borsuk and Grunbaum on lattice polytopes,” Izv. Ross. Akad. Nauk Ser. Mat. 69(3), 81–108 (2005) [Izv.Math. 69 (3), 513–537 (2005)].

    Article  MathSciNet  Google Scholar 

  7. A. Soifer, The Mathematical Coloring Book. Mathematics of Coloring and the Colorful Life of its Creators (Springer, New York, 2009).

    MATH  Google Scholar 

  8. A. M. Raigorodskii, “The Erdős-Hadwiger problem and the chromatic numbers of finite geometric graphs,” Mat. Sb. 196(1), 123–156 (2005) [Sb.Math. 196 (1), 115–146 (2005)].

    Article  MathSciNet  Google Scholar 

  9. E. I. Ponomarenko and A. M. Raigorodskii, “The improvement theorem Frankl-Wilson about number edges hypergraph with prohibitions on intersection,” Dokl. Ross. Akad. Nauk 454(3), 268–269 (2014) [Dokl. Math. 89 (1), 59–60 (2014)].

    MathSciNet  Google Scholar 

  10. E. I. Ponomarenko and A. M. Raigorodskii, “New estimates in the problem of the number of edges in a hypergraph with forbidden intersections,” Problemy Peredachi Informatsii 49(4), 98–104 (2013) [Problems Inform. Transmission 49 (4), 384–390 (2013)].

    MathSciNet  Google Scholar 

  11. L. A. Székely, “Erdős on unit distances and the Szemerédi-Trotter theorems,” in Paul Erdős and his Mathematics, II, Bolyai Soc. Math. Stud. (János BolyaiMath. Soc., Budapest, 2002), Vol. 11, pp. 649–666.

    Google Scholar 

  12. L. I. Bogolyubskii, A. S. Gusev, M. M. Pyaderkin, and A. M. Raigorodskii, “numbers independence and chromatic number randomsubgraphs in some sequences graphs,” Dokl.Ross. Akad. Nauk 457(4), 383–387 (2014) [Dokl.Math. 90 (1), 462–465 (2014)].

    Google Scholar 

  13. L. I. Bogolyubskii, A. S. Gusev, M. M. Pyaderkin, and A. M. Raigorodskii, “Independence numbers and chromatic numbers of random subgraphs of certain distance graphs,” Mat. Sb. (in press).

  14. 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(4), 49–78 (2013) [Sb. Math. 204 (4), 508–538 (2013)].

    Article  MathSciNet  Google Scholar 

  15. J. Balogh, A. Kostochka, and A. Raigorodskii, “Coloring some finite sets in ℝn,” Discuss. Math. Graph Theory 33(1), 25–31 (2013).

    Article  MATH  MathSciNet  Google Scholar 

  16. A. M. Raigorodskii, “On the chromatic numbers of spheres in ℝn,” Combinatorica 32(1), 111–123 (2012).

    Article  MATH  MathSciNet  Google Scholar 

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

    Google Scholar 

  18. A. M. Raigorodskii, “Three lectures on the Borsuk partition problem,” in Surveys in Contemporary Mathematics, London Math. Soc. Lecture Note Ser. (Cambridge Univ. Press, Cambridge, 2007), Vol. 347, pp. 202–247.

    Google Scholar 

  19. A. M. Raigorodskii, “Around Borsuk’s hypothesis,” Sovrem. Mat., Fundam. Napravl. 23, 147–164 (2007) [J. Math. Sci. 154 (4), 604–623 (2008)].

    Google Scholar 

  20. Z. Baranyai, “On the factorization of the complete uniform hypergraph,” in Infinite and Finite Sets (North-Holland, Amsterdam, 1975), Vol. 1, pp. 91–108.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Lomonosov Moscow State University, Moscow, Russia

    V. V. Utkin

Authors
  1. V. V. Utkin
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to V. V. Utkin.

Additional information

Original Russian Text © V. V. Utkin, 2015, published in Matematicheskie Zametki, 2015, Vol. 97, No. 6, pp. 904–916.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Utkin, V.V. Hamiltonian paths in distance graphs. Math Notes 97, 919–929 (2015). https://doi.org/10.1134/S0001434615050260

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S0001434615050260

Keywords

Search

Navigation