Triangulations of the sphere, bitrades and abelian groups

Abstract

Let \(\mathcal{G}\) be a triangulation of the sphere with vertex set V, such that the faces of the triangulation are properly coloured black and white. Motivated by applications in the theory of bitrades, Cavenagh and Wanless defined \(\mathcal{A}_W\) to be the abelian group generated by the set V, with relations r+c+s = 0 for all white triangles with vertices r, c and s. The group \(\mathcal{A}_B\) can be de fined similarly, using black triangles.

The paper shows that \(\mathcal{A}_W\) and \(\mathcal{A}_B\) are isomorphic, thus establishing the truth of a well-known conjecture of Cavenagh and Wanless. Connections are made between the structure of \(\mathcal{A}_W\) and the theory of asymmetric Laplacians of finite directed graphs, and weaker results for orientable surfaces of higher genus are given. The relevance of the group \(\mathcal{A}_W\) to the understanding of the embeddings of a partial latin square in an abelian group is also explained.

This is a preview of subscription content, access via your institution.

References

  1. [1]

    N. Biggs: Algebraic potential theory on graphs, Bull. London Math. Soc. 29 (2007), 641–682.

    Article  MathSciNet  Google Scholar 

  2. [2]

    J. A. Bondy and U. S. R. Murty: Graph Theory, Graduate Texts in Mathematics, 244. Springer, New York, 2008.

    Book  MATH  Google Scholar 

  3. [3]

    P. J. Cameron: Research problems from the BCC22, Discrete Math. 311 (2011), 1074–1083.

    Article  MathSciNet  Google Scholar 

  4. [4]

    N. J. Cavenagh: Embedding 3-homogeneous latin trades into abelian 2-groups, Comentat. Math. Univ. Carolin. 45 (2004), 194–212.

    MathSciNet  Google Scholar 

  5. [5]

    N. J. Cavenagh: The theory and application of latin bitrades: a survey, Math. Slovaca 58 (2008), 691–718.

    Article  MATH  MathSciNet  Google Scholar 

  6. [6]

    N. J. Cavenagh and I. M. Wanless: Latin trades in groups defined on planar triangulations, J. Algebr. Comb. 30 (2009), 323–347.

    Article  MATH  MathSciNet  Google Scholar 

  7. [7]

    Aleš Drápal and N. Cavenagh: Open Problem 8, Open problems from Workshop on latin trades, Prague, 6–10 February 2006. http://www.karlin.mff.cuni.cz/~rozendo/op.html

    Google Scholar 

  8. [8]

    C. J. Colbourn and J. H. Dinitz, (eds.), Second Edition, The CRC Handbook of Combinatorial Designs, Chapman & Hall/CRC Press, Boca Raton (2006).

    Google Scholar 

  9. [9]

    A. Drápal: Hamming distances of groups and quasigroups, Discrete Math. 235 (2001), 189–197.

    Article  MATH  MathSciNet  Google Scholar 

  10. [10]

    Alěs Drápal: On elementary moves that generate all spherical Latin trades, Comment. Math. Univ. Carolin. 50 (2009), no. 4, 477–511.

    MATH  MathSciNet  Google Scholar 

  11. [11]

    Alěs Drápal: Geometrical structure and construction of Latin trades, Adv. Geom. 9 (2009), no. 3, 311–348.

    Article  MATH  MathSciNet  Google Scholar 

  12. [12]

    A. Drápal, C. Hämäläinen and Vítězslav Kala: Latin bitrades, dissections of equilateral triangles, and abelian groups, J. Comb. Des., 18 (2010), 1–24.

    MATH  Google Scholar 

  13. [13]

    Aleš Drápal and Tomáš Kepka: Group modifications of some partial groupoids, Annals of Discr. Math. 18 (1983), 319–332.

    MATH  Google Scholar 

  14. [14]

    Aleš Drápal and Tomáš Kepka: Exchangeable partial groupoids I, Acta Universitatis Carolinae — Mathematica et Physica 24 (1983), 57–72.

    MATH  MathSciNet  Google Scholar 

  15. [15]

    A. D. Forbes, M. J. Grannell and T. S. Griggs: Configurations and trades in Steiner triple systems, Australasian J. Comb. 29 (2004), 75–84.

    MATH  MathSciNet  Google Scholar 

  16. [16]

    P. J. Heawood: On the four-colour map theorem, Quart J. Pure Math. 29 (1898), 270–285.

    MATH  Google Scholar 

  17. [17]

    V. D. Mazurov and E. I. Khukhro (eds), The Kourovka notebook (Unsolved Problems in group theory), 17th edition, Russian Academy of Sciences, Siberian Division, Institute of Mathematics, Novosibirsk, 2010.

    MATH  Google Scholar 

  18. [18]

    J. J. Rotman: An Introduction to the Theory of Groups (3rd edition), Wm. C. Brown Publishers, Dubuque, Iowa, 1988.

    Google Scholar 

  19. [19]

    W. T. Tutte: Graph theory, Addison-Wesley, Reading, MA, 1984.

    MATH  Google Scholar 

  20. [20]

    R. A. Wilson: Graphs, colourings and the four-colour theorem, Oxford University Press, Oxford, 2002.

    MATH  Google Scholar 

Download references

Author information

Affiliations

Authors

Corresponding author

Correspondence to Simon R. Blackburn.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Blackburn, S.R., McCourt, T.A. Triangulations of the sphere, bitrades and abelian groups. Combinatorica 34, 527–546 (2014). https://doi.org/10.1007/s00493-014-2924-7

Download citation

Mathematics Subject Classification (2000)

  • 05C10
  • 05B07
  • 05E99