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Biembeddings of Symmetric \(n\)-Cycle Systems

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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 \(n\ge 4\).

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References

  1. Buratti, M., Del Fra, A.: Existence of cyclic \(k\)-cycle systems of the complete graph. Discrete Math. 261, 113–125 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  3. Franklin, P.: A six color problem. J. Math. Phys. 16, 363–369 (1934)

    Article  Google Scholar 

  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)

  5. Grannell, M.J., Korzhik, V.P.: Nonorientable biembeddings of Steiner triple systems. Discrete Math. 285, 121–126 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  6. Gross, J.L., Alpert, S.R.: The topological theory of current graphs. J. Comb. Theory Ser. B 17, 218–233 (1974)

    Article  MATH  MathSciNet  Google Scholar 

  7. Gross, J.L., Tucker, T.W.: Topological graph theory. Wiley, New York (1987)

    MATH  Google Scholar 

  8. Ringel, G.: Map color theorem. Springer, New York (1974)

    Book  MATH  Google Scholar 

  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)

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Acknowledgments

The authors wish to thank a referee for a very careful reading of the paper and comments which have made the exposition clearer.

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Authors and Affiliations

  1. Department of Mathematics and Statistics, The Open University, Walton Hall, Milton Keynes, MK7 6AA, UK

    Terry S. Griggs

  2. School of Computing and Mathematics, Plymouth University, Drake Circus, Plymouth, PL4 8AA, UK

    Thomas A. McCourt

  3. Heilbronn Institute for Mathematical Research, University of Bristol, University Walk, Bristol, BS8 1TW, UK

    Thomas A. McCourt

Authors
  1. Terry S. Griggs
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  2. Thomas A. McCourt
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Correspondence to Thomas A. McCourt.

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Griggs, T.S., McCourt, T.A. Biembeddings of Symmetric \(n\)-Cycle Systems. Graphs and Combinatorics 32, 147–160 (2016). https://doi.org/10.1007/s00373-015-1538-1

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  • DOI: https://doi.org/10.1007/s00373-015-1538-1

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