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

Quarter-regular biembeddings of Latin squares

We apply a recursive construction for biembeddings of Latin squares to produce a new infinite family of biembeddings of cyclic Latin squares of even side having a high degree of symmetry. Reapplication of the construction yields two further classes of biembeddings. © 2009 Elsevier B.V. All rights reserved