6 research outputs found
Further biembeddings of twofold triple systems
We construct face two-colourable triangulations of the graph 2Kn in an orientable surface; equivalently biembeddings of two twofold triple systems of order n, for all n Ο 16 or 28 (mod 48). The biembeddings come from index 1 current graphs lifted under a group â€n/4 Ă 4
Square Integer Heffter Arrays with Empty Cells
A Heffter array is an matrix with nonzero entries
from such that each row contains filled cells and
each column contains filled cells, every row and column sum to 0, and
no element from appears twice. Heffter arrays are useful in
embedding the complete graph on an orientable surface where the
embedding has the property that each edge borders exactly one cycle and one
cycle. Archdeacon, Boothby and Dinitz proved that these arrays can be
constructed in the case when , i.e. every cell is filled. In this paper we
concentrate on square arrays with empty cells where every row sum and every
column sum is in . We solve most of the instances of this case.Comment: 20 pages, including 2 figure
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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
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 â 1 for small m (m †22), is given. As far as we know, for m â {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