Graphs and Combinatorics

, Volume 32, Issue 1, pp 147–160 | Cite as

Biembeddings of Symmetric \(n\)-Cycle Systems

Original Paper
  • 62 Downloads

Abstract

We construct face two-colourable embeddings of the complete graph \(K_{2n+1}\) in which every face is a cycle of length \(n\); equivalently biembeddings of pairs of symmetric \(n\)-cycle systems. We prove that the necessary and sufficient condition for the existence of such an embedding in an orientable surface is for \(n\ge 3\) to be odd, and in a nonorientable is for \(n\ge 4\).

Keywords

Orientable surface Nonorientable surface Symmetric \(n\)-cycle system Biembedding 

Mathematics Subject Classification

05B30 05C10 

References

  1. 1.
    Buratti, M., Del Fra, A.: Existence of cyclic \(k\)-cycle systems of the complete graph. Discrete Math. 261, 113–125 (2003)MATHMathSciNetCrossRefGoogle Scholar
  2. 2.
    Ellingham, M.N., Stephens, C.: The nonorientable genus of joins of complete graphs with large edgeless graphs. J. Comb. Theory Ser. B. 97, 827–845 (2007)MATHMathSciNetCrossRefGoogle Scholar
  3. 3.
    Franklin, P.: A six color problem. J. Math. Phys. 16, 363–369 (1934)CrossRefGoogle Scholar
  4. 4.
    Grannell, M.J., Griggs, T.S.: Designs and topology, in surveys in combinatorics. In: Hilton, A.J.W., Talbot, J. (Eds.), London Math. Soc. Lecture Note Series 346, pp. 121–174. Cambridge Univ. Press, Cambridge (2007)Google Scholar
  5. 5.
    Grannell, M.J., Korzhik, V.P.: Nonorientable biembeddings of Steiner triple systems. Discrete Math. 285, 121–126 (2004)MATHMathSciNetCrossRefGoogle Scholar
  6. 6.
    Gross, J.L., Alpert, S.R.: The topological theory of current graphs. J. Comb. Theory Ser. B 17, 218–233 (1974)MATHMathSciNetCrossRefGoogle Scholar
  7. 7.
    Gross, J.L., Tucker, T.W.: Topological graph theory. Wiley, New York (1987)MATHGoogle Scholar
  8. 8.
    Ringel, G.: Map color theorem. Springer, New York (1974)MATHCrossRefGoogle Scholar
  9. 9.
    Youngs, J.W.T.: The mystery of the Heawood conjecture, in graph theory and its applications, pp. 17–50. Academic Press, New York (1970)Google Scholar

Copyright information

© Springer Japan 2015

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsThe Open UniversityMilton KeynesUK
  2. 2.School of Computing and MathematicsPlymouth UniversityPlymouthUK
  3. 3.Heilbronn Institute for Mathematical ResearchUniversity of BristolBristolUK

Personalised recommendations