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Graphs and Combinatorics

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

Biembeddings of Symmetric \(n\)-Cycle Systems

  • Terry S. Griggs
  • Thomas A. McCourt
Original Paper
  • 77 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 

Notes

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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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

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