Dihedral Biembeddings and Triangulations by Complete and Complete Tripartite Graphs

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 Kn,n,n and the complete graph Kn 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 Kn and $${n^{2n^2}}$$ in the case of complete tripartite graphs Kn,n,n.

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