Skip to main content
SpringerLink
Log in

Describing Neighborhoods of 5-Vertices in a Class of 3-Polytopes with Minimum Degree 5

  • Published:
Siberian Mathematical Journal Aims and scope Submit manuscript

Abstract

Lebesgue proved in 1940 that each 3-polytope with minimum degree 5 contains a 5-vertex for which the set of degrees of its neighbors is majorized by one of the following sequences

(6, 6, 7, 7, 7), (6, 6, 6, 7, 9), (6, 6, 6, 6, 11)

(5, 6, 7, 7, 8), (5, 6, 6, 7, 12), (5, 6, 6, 8, 10), (5, 6, 6, 6, 17)

(5, 5, 7, 7, 13), (5, 5, 7, 8, 10), (5, 5, 6, 7, 27), (5, 5, 6, 6,∞), (5, 5, 6, 8, 15), (5, 5, 6, 9, 11)

(5, 5, 5, 7, 41), (5, 5, 5, 8, 23), (5, 5, 5, 9, 17), (5, 5, 5, 10, 14), (5, 5, 5, 11, 13).

We prove that each 3-polytope with minimum degree 5 without vertices of degree from 7 to 10 contains a 5-vertex whose set of degrees of its neighbors is majorized by one of the following sequences: (5, 6, 6, 5, ∞), (5, 6, 6, 6, 15), and (6, 6, 6, 6, 6), where all parameters are tight.

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. Steinitz E., “Polyeder und Raumeinteilungen,” Enzykl. Math. Wiss. (Geometrie), vol. 3AB, no. 12, 1–139 (1922).

    Google Scholar 

  2. Wernicke P., “Über den kartographischen Vierfarbensatz,” Math. Ann., Bd 58, 413–426 (1904).

    Article  MathSciNet  MATH  Google Scholar 

  3. Franklin Ph., “The four color problem,” Amer. J. Math., vol. 44, no. 3, 225–236 (1922).

    Article  MathSciNet  MATH  Google Scholar 

  4. Borodin O. V. and Ivanova A. O., “An analogue of Franklin’s theorem,” Discrete Math., vol. 339, no. 10, 2553–2556 (2016).

    Article  MathSciNet  MATH  Google Scholar 

  5. Lebesgue H., “Quelques conséquences simples de la formule d’Euler,” J. Math. Pures Appl., vol. 19, 27–43 (1940).

    MathSciNet  MATH  Google Scholar 

  6. Borodin O. V., “Colorings of plane graphs: a survey,” Discrete Math., vol. 313, no. 4, 517–539 (2013).

    Article  MathSciNet  MATH  Google Scholar 

  7. Ore O. and Plummer M. D., “Cyclic coloration of plane graphs,” in: Recent Progress in Combinatorics, Academic Press, New York, 1969, 287–293.

    Google Scholar 

  8. Kotzig A., “From the theory of Eulerian polyhedra” (Russian), Mat. Eas. SAV (Math. Slovaca), vol. 13, 20–34 (1963).

    MATH  Google Scholar 

  9. Borodin O. V., “Solution of Kotzig’s and Grünbaum’s problems on the separability of a cycle in a planar graph” (Russian), Mat. Zametki, vol. 46, no. 5, 9–12 (1989).

    MathSciNet  Google Scholar 

  10. Grünbaum B., “Polytopal graphs. Studies in graph theory. P. II,” Stud. Math., Math. Assoc. Amer., vol. 12, 201–224 (1975).

    Google Scholar 

  11. Jendrol’ S. and Madaras T., “On light subgraphs in plane graphs of minimal degree five,” Discuss. Math. Graph Theory, vol. 16, 207–217 (1996).

    Article  MathSciNet  MATH  Google Scholar 

  12. Borodin O. V. and Ivanova A. O., “Describing 4-stars at 5-vertices in normal plane maps with minimum degree 5,” Discrete Math., vol. 313, no. 17, 1710–1714 (2013).

    Article  MathSciNet  MATH  Google Scholar 

  13. Borodin O. V. and Woodall D. R., “Short cycles of low weight in normal plane maps with minimum degree 5,” Discuss. Math. Graph Theory, vol. 8, no. 2, 159–164 (1998).

    Article  MathSciNet  MATH  Google Scholar 

  14. Borodin O. V., Ivanova A. O., and Jensen T. R., “5-Stars of low weight in normal plane maps with minimum degree 5,” Discuss. Math. Graph Theory, vol. 34, no. 3, 539–546 (2014).

    Article  MathSciNet  MATH  Google Scholar 

  15. Borodin O. V. and Ivanova A. O., “Light and low 5-stars in normal plane maps with minimum degree 5,” Sib. Math. J., vol. 57, no. 3, 596–602 (2016).

    Article  MathSciNet  MATH  Google Scholar 

  16. Borodin O. V. and Ivanova A. O., “Light neighborhoods of 5-vertices in 3-polytopes with minimum degree 5,” Sib. Elektron. Math. Rep., vol. 13, 584–591 (2016).

    MathSciNet  MATH  Google Scholar 

  17. Borodin O. V. and Ivanova A. O., “Low 5-stars in normal plane maps with minimum degree 5,” Discrete Math., vol. 340, no. 2, 18–22 (2017).

    Article  MathSciNet  MATH  Google Scholar 

  18. Borodin O. V. and Ivanova A. O., “On light neighborhoods of 5-vertices in 3-polytopes with minimum degree 5,” Discrete Math., vol. 340, no. 9, 2234–2242 (2017).

    Article  MathSciNet  MATH  Google Scholar 

  19. Borodin O. V., Ivanova A. O., and Nikiforov D.V., “Low minor 5-stars in 3-polytopes with minimum degree 5 and no 6-vertices,” Discrete Math., vol. 340, no. 7, 1612–1616 (2017).

    Article  MathSciNet  MATH  Google Scholar 

  20. Jendrol’ S. and Madaras T., “Note on an existence of small degree vertices with at most one big degree neighbour in planar graphs,” Tatra Mt. Math. Publ., vol. 30, 149–153 (2005).

    MathSciNet  MATH  Google Scholar 

  21. Ivanova A. O. and Nikiforov D. V., “The structure of neighborhoods of 5-vertices in plane triangulation with minimum degree 5” (Russian), Mat. Zam. YaSU, vol. 20, no. 2, 66–78 (2013).

    MATH  Google Scholar 

  22. Ivanova A. O. and Nikiforov D. V., “Combinatorial structure of triangulated 3-polytopes with minimum degree 5,” in: Proceedings of the Scientific Conference of Students, Graduate Students, and Young Researchers, XVII and XVIII Lavrent'ev’s Reading, Yakutsk, Kirov, International Center for Research Project, 2015, 2015, 22–27.

    Google Scholar 

  23. Nikiforov D. V., “The structure of neighborhoods of 5-vertices in normal plane maps with minimum degree 5” (Russian), Mat. Zam. YaSU, vol. 23, no. 1, 56–66 (2016).

    MATH  Google Scholar 

  24. Borodin O. V., Ivanova A. O., and Kazak O. N., “Describing neighborhoods of 5-vertices in 3-polytopes with minimum degree 5 and without vertices of degrees from 7 to 11,” Discuss. Math. Graph Theory; DOI:10.7151/dmgt.2024.

Download references

Author information

Authors and Affiliations

  1. Sobolev Institute of Mathematics, Novosibirsk, Russia

    O. V. Borodin, A. O. Ivanova & D. V. Nikiforov

Authors
  1. O. V. Borodin
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. A. O. Ivanova
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. D. V. Nikiforov
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to O. V. Borodin.

Additional information

Original Russian Text Copyright © 2018 Borodin O.V., Ivanova A.O., and Nikiforov D.V.

The authors were funded by the Russian Science Foundation (Grant 16–11–10054).

Translated from Sibirskiĭ Matematicheskiĭ Zhurnal, Vol. 59, No. 1, pp. 56–64, January–February, 2018; DOI: 10.17377/smzh.2018.59.105

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Borodin, O.V., Ivanova, A.O. & Nikiforov, D.V. Describing Neighborhoods of 5-Vertices in a Class of 3-Polytopes with Minimum Degree 5. Sib Math J 59, 43–49 (2018). https://doi.org/10.1134/S0037446618010056

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S0037446618010056

Keywords

Search

Navigation