Skip to main content
SpringerLink
Log in

Note on 4-Coloring 6-Regular Triangulations on the Torus

  • Published:
Annals of Combinatorics Aims and scope Submit manuscript

Abstract

Altshuler (Discrete Math 4(3):201–217, 1973) characterized the 6-regular triangulations on the torus to be precisely those that are obtained from a regular triangulation of the \(r \times s\) toroidal grid where the vertices in the first and last column are connected by a shift of t vertices. Such a graph is denoted T(r, s, t). Collins and Hutchinson (Graph colouring and applications. CRM proceedings and lecture notes, vol 23. American Mathematical Society, Providence, pp 21–34, 1999) classified the 4-colorable graphs T(r, s, t) with \(r, s \ge 3\). In this paper, we point out a gap in their classification and show how it can be fixed. Combined with the classification of the 4-colorable graphs T(1, s, t) by Yeh and Zhu (Discrete Math 273(1–3):261–274, 2003), this completes the characterization of the colorability of all the 6-regular triangulations on the torus.

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

Similar content being viewed by others

Availability of Data and Materials

Not applicable.

References

  1. Albertson, M.O., Hutchinson, J.P.: On six-chromatic toroidal graphs. Proc. Lond. Math. Soc. Third Ser. 41(3), 533–556 (1980). https://doi.org/10.1112/plms/s3-41.3.533. MR0591654, Zbl 0394.05018.

  2. Altshuler, A.: Hamiltonian circuits in some maps on the torus. Discret. Math. 1(4), 299–314 (1972). https://doi.org/10.1016/0012-365X(72)90037-4. MR0297597, Zbl 0226.05109.

  3. Altshuler, A.: Construction and enumeration of regular maps on the torus. Discret. Math. 4(3), 201–217 (1973). https://doi.org/10.1016/S0012-365X(73)80002-0. MR0321797, Zbl 0253.05117.

  4. Balachandran, N., Sankarnarayanan, B.: The choice number versus the chromatic number for graphs embeddable on orientable surfaces. Electron. J. Comb. 28(4), #P4.50 (2021). https://doi.org/10.37236/10263.

  5. Brooks, R.L.: On colouring the nodes of a network. Proc. Camb. Philos. Soc. 37(2), 194–197 (1941). https://doi.org/10.1017/S030500410002168X. MR0012236, Zbl 0027.26403.

  6. Collins, K.L., Hutchinson, J.P.: Four-coloring six-regular graphs on the torus. In: Hansen, P., Marcotte, O. (eds.) Graph Colouring and Applications, CRM Proc. Lect. Notes, vol. 23, pp. 21–34. Am. Math. Soc., Providence, R. I. (1999). https://doi.org/10.1090/crmp/023/02. MR1723634, Zbl 0944.05044.

  7. Dirac, G.A.: Map-colour theorems. Can. J. Math. 4, 480–490 (1952). https://doi.org/10.4153/cjm-1952-043-9. MR0050869, Zbl 0047.42203.

  8. Heawood, P.J.: Map-colour theorem. Q. J. Pure Appl. Math. 24(96), 332–338 (1890). Available at http://resolver.sub.uni-goettingen.de/purl?PPN600494829_0024. JFM 22.0562.02.

  9. Negami, S.: Uniqueness and faithfulness of embedding of toroidal graphs. Discret. Math. 44(2), 161–180 (1983). https://doi.org/10.1016/0012-365X(83)90057-2. MR0689809, Zbl 0508.05033.

  10. Thomassen, C.: Tilings of the torus and the Klein bottle and vertex-transitive graphs on a fixed surface. Trans. Am. Math. Soc. 323(2), 605–635 (1991). https://doi.org/10.2307/2001547. MR1040045, Zbl 0722.05031.

  11. Yeh, H.-G., Zhu, X.: \(4\)-colorable \(6\)-regular toroidal graphs. Discret. Math. 273(1–3), 261–274 (2003). https://doi.org/10.1016/S0012-365X(03)00242-5. MR2025955, Zbl 1034.05024.

Download references

Acknowledgements

The author is grateful to his advisor Niranjan Balachandran for helpful comments and discussions. The author also wishes to thank an anonymous reviewer for a careful reading and helpful suggestions for improvement in the presentation. This work is supported by a PhD fellowship received from the National Board for Higher Mathematics (NBHM), Department of Atomic Energy (DAE), Govt. of India.

Funding

This work is supported by a PhD fellowship received from the National Board for Higher Mathematics (NBHM), Department of Atomic Energy (DAE), Govt. of India.

Author information

Authors and Affiliations

  1. Department of Mathematics, Indian Institute of Technology Bombay, Powai, Mumbai, Maharashtra, 400076, India

    Brahadeesh Sankarnarayanan

Authors
  1. Brahadeesh Sankarnarayanan
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Brahadeesh Sankarnarayanan.

Ethics declarations

Conflict of Interest

The author has no competing interests to declare that are relevant to the content of this article.

Code Availability

Not applicable.

Additional information

Communicated by Kolja Knauer.

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Sankarnarayanan, B. Note on 4-Coloring 6-Regular Triangulations on the Torus. Ann. Comb. 26, 559–569 (2022). https://doi.org/10.1007/s00026-022-00573-8

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00026-022-00573-8

Keywords

Mathematics Subject Classification

Search

Navigation