Abstract. In series of papers we proved that for some (almost) linear classes of orders n, there are at least anbn2 nonisomorphic face 2-colourable triangular embeddings of Kn, where a,b>0. These triangulations are obtained using face 2-colourable triangulations by Km,m,m. Since each colour class of a triangulation by Km,m,m corresponds to a Latin square, we can consider these later maps as biembeddings of Latin squares.
Earlier we proved that among Latin squares embedded in this way we have all the Cayley tables of Abelian groups, with the sole exception of C22. Here we prove that every Cayley table of an Abelian group except C2, C22 and C4 is in such a biembedding, in which the second square has a transversal. This result not only shows an interesting "topological" feature of Latin squares, but also can be (and in a recent paper is) used in a certain recursive construction to produce "many" triangulations by Km,m,m and consequently by Kn.