12 research outputs found
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
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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
Globally simple Heffter arrays and orthogonal cyclic cycle decompositions
In this paper we introduce a particular class of Heffter arrays, called
globally simple Heffter arrays, whose existence gives at once orthogonal cyclic
cycle decompositions of the complete graph and of the cocktail party graph. In
particular we provide explicit constructions of such decompositions for cycles
of length . Furthermore, starting from our Heffter arrays we also
obtain biembeddings of two -cycle decompositions on orientable surfaces.Comment: The present version also considers the problem of biembedding
Self-embeddings of Hamming Steiner triple systems of small order and APN permutations
The classification, up to isomorphism, of all self-embedding monomial power permutations of Hamming Steiner triple systems of order n = 2 m − 1 for small m (m ≤ 22), is given. As far as we know, for m ∈ {5, 7, 11, 13, 17, 19}, all given self-embeddings in closed surfaces are new. Moreover, they are cyclic for all m and nonorientable at least for all m ≤ 19. For any non prime m, the nonexistence of such self-embeddings in a closed surface is proven. The rotation line spectrum for self-embeddings of Hamming Steiner triple systems in pseudosurfaces with pinch points as an invariant to distinguish APN permutations or, in general, to classify permutations, is also proposed. This invariant applied to APN monomial power permutations gives a classification which coincides with the classification of such permutations via CCZ-equivalence, at least up to m ≤ 17