Abstract
We construct biembeddings of some Latin squares which are Cayley tables of dihedral groups. These facilitate the construction of \({n^{an^2}}\) nonisomorphic face 2-colourable triangular embeddings of the complete tripartite graph K n,n,n and the complete graph K n for linear classes of values of n and suitable constants a. Previously the best known lower bounds for the number of such embeddings that are applicable to linear classes of values of n were of the form \({2^{an^2}.}\) We remark that trivial upper bounds are \({n^{n^2/3}}\) in the case of complete graphs K n and \({n^{2n^2}}\) in the case of complete tripartite graphs K n,n,n .
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Grannell, M.J., Knor, M. Dihedral Biembeddings and Triangulations by Complete and Complete Tripartite Graphs. Graphs and Combinatorics 29, 921–932 (2013). https://doi.org/10.1007/s00373-012-1163-1
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DOI: https://doi.org/10.1007/s00373-012-1163-1