Skip to main content
SpringerLink
Log in

Combinatorial Structure of Faces in Triangulations on Surfaces

  • Published:
Siberian Mathematical Journal Aims and scope Submit manuscript

Abstract

The degree \( d(x) \) of a vertex or face \( x \) in a graph \( G \) on the plane or other orientable surface is the number of incident edges. A face \( f=v_{1}\ldots v_{d(f)} \) is of type \( (k_{1},k_{2},\dots) \) if \( d(v_{i})\leq k_{i} \) whenever \( 1\leq i\leq d(f) \). We denote the minimum vertex-degree of \( G \) by \( \delta \). The purpose of our paper is to prove that every triangulation with \( \delta\geq 4 \) of the torus, as well as of large enough such a triangulation of any fixed orientable surface of higher genus has a face of one of the types \( (4,4,\infty) \), \( (4,6,12) \), \( (4,8,8) \), \( (5,5,8) \), \( (5,6,7) \), or \( (6,6,6) \), where all parameters are best possible.

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

Fig. 1
Fig. 4
Fig. 5

Similar content being viewed by others

References

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

    MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

  4. Kotzig A., “From the theory of Eulerian polyhedra,” Mat. Čas., vol. 13, 20–31 (1963) [Russian].

    MATH  Google Scholar 

  5. Borodin O. V., “Solution of Kotzig’s and Grübaum’s problems on the separability of a cycle in a planar graph,” Math. Notes, vol. 46, no. 5, 835–837 (1989).

    Article  MathSciNet  Google Scholar 

  6. Grünbaum B., “Polytopal graphs,” in: Studies in Graph Theory. Part II. Fulkerson D., ed., MAA (1975), 201–224 (MAA Studies in Mathematics; Vol. 12).

  7. Plummer M. D., “On the cyclic connectivity of planar graphs,” in: Graph Theory and Applications. Proceedings of the Conference at Western Michigan University, May 10–13, 1972 (1972), 235–242 (Lecture Notes Math.; Vol. 303).

  8. Borodin O. V. and Ivanova A. O., “New results about the structure of plane graphs: a survey,” AIP Conference Proceedings, vol. 1907, no. 1, 030051 (2017).

    Article  Google Scholar 

  9. Jendrol’ S. and Voss H.-J., “Light subgraphs of graphs embedded in the plane—a survey,” Discrete Math., vol. 313, no. 4, 406–421 (2013).

    Article  MathSciNet  Google Scholar 

  10. Borodin O. V., “Triangulated 3-polytopes without faces of low weight,” Discrete Math., vol. 186, no. 1, 281–285 (1998).

    Article  MathSciNet  Google Scholar 

  11. Borodin O. V., “An improvement of Lebesgue’s theorem on the structure of minor faces of 3-polytopes,” Diskretn. Anal. Issled. Oper., vol. 9, no. 3, 29–39 (2002) [Russian].

    MathSciNet  MATH  Google Scholar 

  12. Borodin O. V. and Ivanova A. O., “Describing 3-faces in normal plane maps with minimum degree 4,” Discrete Math., vol. 313, no. 23, 2841–2847 (2013).

    Article  MathSciNet  Google Scholar 

  13. Horňák M. and Jendrol’ S., “Unavoidable sets of face types for planar maps,” Discuss. Math. Graph Theory, vol. 16, no. 2, 123–142 (1996).

    Article  MathSciNet  Google Scholar 

  14. Jendrol’ S., “Triangles with restricted degrees of their boundary vertices in plane triangulations,” Discrete Math., vol. 196, no. 1, 177–196 (1999).

    Article  MathSciNet  Google Scholar 

  15. Mohar B., Škrekovski R., and Voss H.-J., “Light subgraphs in planar graphs of minimum degree 4 and edge-degree 9,” J. Graph Theory, vol. 44, no. 4, 261–295 (2003).

    Article  MathSciNet  Google Scholar 

  16. Borodin O. V., Ivanova A. O., and Kostochka A. V., “Describing faces in plane triangulations,” Discrete Math., vol. 319, no. 1, 47–61 (2014).

    Article  MathSciNet  Google Scholar 

  17. Borodin O. V. and Ivanova A. O., “An improvement of Lebesgue’s description of edges in \( 3 \)-polytopes and faces in plane quadrangulations,” Discrete Math., vol. 342, no. 6, 1820–1827 (2019).

    Article  MathSciNet  Google Scholar 

  18. Avgustinovich S. V. and Borodin O. V., “Neighborhoods of edges in normal maps,” Diskret. Anal. Issled. Oper., vol. 2, no. 3, 3–9 (1995) (English translation: Oper. Res. Discrete Anal., vol. 391, 17–22 (1997)).

    MathSciNet  MATH  Google Scholar 

  19. Borodin O. V. and Ivanova A. O., “Tight description of faces in torus triangulations with minimum degree 5,” Siberian Electron. Math. Res., vol. 18, no. 2, 1475–1481 (2021).

    MathSciNet  MATH  Google Scholar 

Download references

Funding

Borodin was supported by the Ministry of Science and Higher Education of the Russian Federation (Grant FWNF–2022–0017). Ivanova was supported by the Ministry of Science and Higher Education of the Russian Federation (Grant FSRG–2020–0006).

Author information

Authors and Affiliations

  1. Sobolev Institute of Mathematics, Novosibirsk, Russia

    O. V. Borodin & A. O. Ivanova

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

Corresponding author

Correspondence to O. V. Borodin.

Additional information

Translated from Sibirskii Matematicheskii Zhurnal, 2022, Vol. 63, No. 4, pp. 796–804. https://doi.org/10.33048/smzh.2022.63.406

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. Combinatorial Structure of Faces in Triangulations on Surfaces. Sib Math J 63, 662–669 (2022). https://doi.org/10.1134/S0037446622040061

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S0037446622040061

Keywords

UDC

Search

Navigation