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Three recursive constructions are presented; two deal with embeddings of complete graphs and one with embeddings of complete tripartite graphs. All three facilitate the construction of 2 an 2 −o(n 2 ) non-isomorphic face 2-colourable triangulations of K n and K n,n,n in orientable and non-orientable surfaces for values of n lying in certain residue classes and for appropriate constants a.

Formulae for the numbers of two, three, and four-line configurations in a Steiner triple system of order u, STS(u), are given. While the formulae for two and three-line configurations depend only on u , the same is true for only 5 of the 16 four-line configurations. For the other 11 and fixed u, the number of occurrences of any one of them, in particular the Pasch configuration, determines the number of occurrences of all the others. 0 1995 John Wiley & SOW, Inc.

AFour methods for constructing anti-Pasch Steiner triple systems are developed. The first generalises a construction of Stinson and Wei to obtain a general singular direct product construction. The second generalises the Bose construction. The third employs a construction due to Lu. The fourth employs Wilsontype inflation techniques using Latin squares having no subsquares of order 2. As a consequence of these constructions we are able to produce anti-Pasch systems of order for 1 or 7 (mod 18), for 49 (mod 72), and for many other values of .

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

We show that an anti-Pasch Steiner triple system of order v exists for v ≡ 1 or 3 (mod 6), apart from v = 7 and 13.

We prove that, for a certain positive constant a and for an infinite set of values of n, the number of nonisomorphic triangular embeddings of the complete graph K n is at least n an 2 . A similar lower bound is also given, for an infinite set of values of n, on the number of nonisomorphic triangular embeddings of the complete regular tripartite graph K n,n,n .

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