Abstract. In every face 2-colourable triangulation of K_t,t,t. each of the two colour classes determines a Latin square. In previous papers we found for every Abelian group G a triangulation such that one of the squares is the Cayley table of G and the other has a transversal. Such embeddings are suitable for constructing many face 2-colourable triangulations of K_z.

In recent paper we generalized this construction for many metacyclic groups. Though at the moment we are not able to do the construction for every metacyclic group, we have sufficiently many of them to prove that there are at least z^z2(a-o(1)) face 2-colourable orientable triangulations of K_z for almost linear class of z's. We remark that in previous papers we achieved analogous bound but for substantially smaller class of z's.