Two Notes on Maps and Surface Symmetry

  • Thomas W. TuckerEmail author
Part of the Fields Institute Communications book series (FIC, volume 70)


The first note of this paper determines for which g the orientable surface of genus g can be embedded in euclidean 3-space so as to have prismatic, cubical/octahedral, tetrahedral, or icosahedral/dodecahedral symmetry. The second note proves, through entirely elementary methods, that the clique number of the graph underlying a regular map is m = 2, 3, 4, 6; for m = 6 the map must be non-orientable and for m = 4, 6 the graph has a K m factorization. Here a regular map is one having maximal symmetry: reflections in all edges and full rotational symmetry about every vertex, edge and face.


Riemann-Hurwitz equation Regular map Clique 

Subject Classifications

05C10 57M15 57M60 


  1. 1.
    Biggs, N.L.: Algebraic Graph Theory. Cambridge Tracts in Mathematics, vol. 67. Cambridge University Press, London (1974)Google Scholar
  2. 2.
    Cavendish, W., Conway, J.H.: Symmetrically bordered surfaces. Am. Math. Mon. 117, 571–580 (2010)CrossRefzbMATHMathSciNetGoogle Scholar
  3. 3.
    Conder, M., Širàň, J., Tucker, T.: The genera, reflexibility, and simplicity of regular maps. J. Eur. Math. Soc. 12, 343–364 (2010)CrossRefzbMATHGoogle Scholar
  4. 4.
    Coxeter, H.S.M., Moser, W.O.J.: Generators and Relations for Discrete Groups, 4th edn. Springer, Berlin (1984)Google Scholar
  5. 5.
    Godfrey, D., Martinez, D. (based on an idea of Tomaž Pisanski): “Tucker’s Group of Genus Two”, sculpture, Technical Museum of Slovenia, Bistra, SloveniaGoogle Scholar
  6. 6.
    Gross, J.L., Tucker, T.W.: Topological Graph Theory, 351 pp. Wiley-Interscience, New York (1987) + xv (Dover paperback, with new foreword and additional references, 2001)Google Scholar
  7. 7.
    James, L.D., Jones, G.A.: Regular orientable imbeddings of complete graphs. J. Comb. Theory Ser. B 39, 353–367 (1985)CrossRefzbMATHMathSciNetGoogle Scholar
  8. 8.
    Kulkarni, R.S.: Symmetries of surfaces. Topology 26, 195–203 (1987)CrossRefzbMATHGoogle Scholar
  9. 9.
    Leopold, U., Tucker, T.: Rotational symmetry of immersed surfaces in S 3 (in preparation)Google Scholar
  10. 10.
    O’Sullivan, C., Weaver, A.: A diophantine frobenius problem related to Riemann surfaces. Glasg. Math. J. 53, 501–522 (2011)CrossRefzbMATHMathSciNetGoogle Scholar
  11. 11.
    Širàň, J., Tucker, T.: Symmetric maps. In: Beineke, L., Wilson, R.J. (eds.) Topics in Topological Graph Theory, pp. 225–244. Cambridge University Press, Cambridge (2009)Google Scholar
  12. 12.
    Tucker, T.: Symmetry of surfaces in 3-space (in preparation)Google Scholar
  13. 13.
    Wilson, S.E.: Cantankerous maps and rotary embeddings of K n. J. Comb. Theory Ser. B 47, 262–279 (1989)CrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  1. 1.Department of MathematicsColgate UniversityHamiltonUSA

Personalised recommendations