Biembeddings of Latin Squares and Hamiltonian Decompositions

Abstract

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

Third-Regular Bi-Embeddings of Latin Squares

“…Furthermore, as proved in [5], the orientability of such a triangular embedding is equivalent to face 2-colourability. By selecting one of the three sets of the tripartition of K n,n,n , and by deleting these vertices and the edges incident with them, one may obtain from T n a regular Hamiltonian embedding of K n,n in an orientable surface.…”

On Parity Vectors of Latin Squares

“…In the context of non-existence results, some small Latin squares do not appear in any biembeddings [5]. In [11], the present authors introduced the concept of the parity vector of a main class of Latin squares.…”

“…A triangular embedding of a complete regular tripartite graph K n,n,n in a surface is face two-colourable if and only if the surface is orientable [5]. In this case, the faces of each colour class can be regarded as the triples of a transversal design T D(3, n), of order n and block size 3.…”

“…An infinite class of biembeddings of Latin squares representing the Cayley tables of cyclic groups of order n is known for all n ≥ 2. This is the family of regular biembeddings constructed using a voltage graph based on a dipole with n parallel edges embedded in a sphere [12], or alternatively directly from the Latin squares defined by C n (i, j) = i + j (mod n), and C ′ n (i, j) = i + j − 1 (mod n) [5]. A regular biembedding of a Latin square of side n has the greatest possible symmetry, with full automorphism group of order 12n 2 , the maximum possible value.…”

“…Recently the present authors [3] constructed another family of biembeddings of the Latin squares representing the Cayley tables of cyclic groups of order 2n, n ≥ 3, also with a high degree of symmetry. Enumeration results for biembeddings of Latin squares of side 3 to 7 are given in [5] and for groups of order 8 in [8]. More recently still, in [9] it has been shown that with the single exception of the group C 2 2 , the Cayley table of each Abelian group appears in some biembedding.…”