On parity vectors of Latin squares. (English) Zbl 1231.05034
Summary: The parity vectors of two Latin squares of the same side \(n\) provide a necessary condition for the two squares to be biembeddable in an orientable surface. We investigate constraints on the parity vector of a Latin square resulting from structural properties of the square, and show how the parity vector of a direct product may be obtained from the parity vectors of the constituent factors. Parity vectors for Cayley tables of all Abelian groups, some non-Abelian groups, Steiner quasigroups and Steiner loops are determined. Finally, we give a lower bound on the number of main classes of Latin squares of side \(n\) that admit no self-embeddings.
MSC:
| 05B15 | Orthogonal arrays, Latin squares, Room squares |
| 05C10 | Planar graphs; geometric and topological aspects of graph theory |
Keywords:
Latin square; orientable surface; biembedding; parity vector; group; Steiner quasigroup; Steiner loopReferences:
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