Abstract. Face 2-colourable triangulations of complete tripartite graphs K n,n,n correspond to biembeddings of Latin squares. Up to isomorphism, we give all such embeddings for n = 3, 4, 5 and 6, and we summarize the corresponding results for n = 7. Closely related to these are Hamiltonian decompositions of complete bipartite directed graphs K * n,n , and we also give computational results for these in the cases n = 3, 4, 5 and 6.2000 Mathematics Subject Classification. 05B15, 05C10.
Introduction.A number of recent papers [2, 5, 6] have dealt with biembeddings of pairs of Steiner triple systems (STSs) in both orientable and nonorientable surfaces. Such a biembedding corresponds to a face 2-colourable triangulation of a complete graph K n . The vertices of the graph form the points of the Steiner triple systems and the triangular faces in each of the two colour classes respectively form the triples of each system. We here recall that an STS(n) may be formally defined as an ordered pair (V, B), where V is an n-element set (the points) and B is a set of 3-element subsets of V (the triples), such that every 2-element subset of V appears in precisely one triple. Such systems are known to exist if and only if n ≡ 1 or 3 (mod 6). We say that two STS(n)s are biembedded in a surface if there is a face 2-colourable triangulation of K n in which the face sets forming the two colour classes give copies of the two systems. We will take the colour classes of face 2-colourable embeddings to be black and white.One may consider embeddings which involve other types of combinatorial design. Embeddings of complete tripartite graphs are discussed in [7,11] and form a useful tool in recursive constructions for biembeddings of Steiner triple systems. A face 2-colourable triangulation of the complete tripartite graph K n,n,n may be considered as a biembedding of a pair of transversal designs TD(3, n); such a design comprises an ordered triple (V, G, B), where V is a 3n-element set (the points), G is a partition of V into three disjoint sets (the groups) each of cardinality n, and B is a set of 3-element subsets of V (the triples), such that every unordered pair of elements from V is either contained in precisely one triple or one group, but not both. As with STSs, the vertices of the embedded graph K n,n,n form the points of the designs, the tripartition