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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...
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