Enumeration of self-orthogonal Latin squares
نویسنده
چکیده
The enumeration of self-orthogonal Latin squares (SOLS) of a given order seems to be an open problem in the literature on combinatorial designs. The existence of at least one SOLS is guaranteed for any order except 2, 3 and 6, but it is not known how many of these squares of a given order exist. In this talk we present enumeration tables of unequal SOLS, idempotent SOLS, isomorphism classes of SOLS and isotopy classes of SOLS. The isotopy classes are enumerated by an (almost) exhaustive computerised tree search which generates a representative from each isotopy class, whereafter the automorphism groups of these representatives are used together with results from abstract algebra in order to enumerate unequal SOLS, idempotent SOLS and isomorphism classes of SOLS. Finally, the results are validated by using an alternative computerised tree search method for all four classes of SOLS. The results appear as sequences A160365, A160366, A160367 and A160368 in Sloane’s Online Encyclopedia of Integer Sequences [1].
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A graph-theoretic proof of the non-existence of self-orthogonal Latin squares of order 6
The non-existence of a pair of mutually orthogonal Latin squares of order six is a well-known result in the theory of combinatorial designs. It was conjectured by Euler in 1782 and was first proved by Tarry [4] in 1900 by means of an exhaustive enumeration of equivalence classes of Latin squares of order six. Various further proofs have since been given [1, 2, 3, 5], but these proofs generally ...
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