Abstract. We consider triangular embeddings of complete tripartite graphs Kn,n,n. If the embedding is face 2-colourable, then the colour classes form Latin squares. Though many of these embeddings were constructed before, only a few Latin squares were distinguished. Here we show that, whenever L is a Latin square formed by the Cayley table of an Abelian group of order n, then it appears, as a colour class, in face 2-coulorable embedding of Kn,n,n.

We remark that the embeddings of Kn,n,n are used to construct "many" nonisomorphic triangulations by complete graphs, which is a problem related to Heawood map colour theorem.