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Keywords:

  • Steiner triple system;
  • Heffter's difference problem;
  • topological embeddings of complete graphs

Abstract

A cyclic face 2-colourable triangulation of the complete graph Kn in an orientable surface exists for n ≡ 7 (mod 12). Such a triangulation corresponds to a cyclic bi-embedding of a pair of Steiner triple systems of order n, the triples being defined by the faces in each of the two colour classes. We investigate in the general case the production of such bi-embeddings from solutions to Heffter's first difference problem and appropriately labelled current graphs. For n = 19 and n = 31 we give a complete explanation for those pairs of Steiner triple systems which do not admit a cyclic bi-embedding and we show how all non-isomorphic solutions may be identified. For n = 43 we describe the structures of all possible current graphs and give a more detailed analysis in the case of the Heawood graph. © 2002 Wiley Periodicals, Inc. J Combin Designs 10: 92–110, 2002; DOI 10.1002/jcd.10001