On biembedding an idempotent latin square with its transpose

Let L be an idempotent Latin square of side n, thought of as a set of ordered triples (i, j, k) where L(I, j) = k. Let I be the set of triples (i, I, i). We consider the problem of biembedding the triples of L\I, with the triples of L'\ I, where L' is the transpose of L, in an orientable surface. We construct such embeddings for all doubly even values of n

Combinatorial Embeddings and Representations

Topological embeddings of complete graphs and complete multipartite graphs give rise to combinatorial designs when the faces of the embeddings are triangles. In this case, the blocks of the design correspond to the triangular faces of the embedding. These designs include Steiner, twofold and Mendelsohn triple systems, as well as Latin squares. We look at construction methods, structural properties and other problems concerning these cases. In addition, we look at graph representations by Steiner triple systems and by combinatorial embeddings. This is closely related to finding independent sets in triple systems. We examine which graphs can be represented in Steiner triple systems and combinatorial embeddings of small orders and give several bounds including a bound on the order of Steiner triple systems that are guaranteed to represent all graphs of a given maximum degree. Finally, we provide an enumeration of graphs of up to six edges representable by Steiner triple systems

Biembeddings of Latin squares obtained from a voltage construction

We investigate a voltage construction for face $2$-colourable triangulations by $K_{n,n,n}$ from the viewpoint of the underlying Latin squares. We prove that if the vertices are relabelled so that one of the Latin squares is exactly the Cayley table $C_n$ of the group $\mathbb Z_n$, then the other square can be obtained from $C_n$ by a cyclic permutation of row, column or entry identifiers, and we identify these cyclic permutations. As an application, we improve the previously known lower bound for the number of nonisomorphic triangulations by $K_{n,n,n}$ obtained from the voltage construction in the case when $n$ is a prime number