Expressions involving the product of the permanent with the (n − 1)st power of the determinant of a matrix of indeterminates, and of (0,1)-matrices, are shown to be related to an extension to odd dimensions of the Alon-Tarsi Latin Square Conjecture, first stated by Zappa. These yield an alternative proof of a theorem of Drisko, stating that the extended conjecture holds for Latin squares of odd...

A 1-factorisation of a graph is perfect if the union of any two of its 1-factors is a Hamiltonian cycle. Let n=p for an odd prime p. We construct a family of (p−1)/2 non-isomorphic perfect 1-factorisations of Kn, n. Equivalently, we construct pan-Hamiltonian Latin squares of order n. A Latin square is pan-Hamiltonian if the permutation defined by any row relative to any other row is a single cy...

To protect memories against errors, error correction codes (ECCs) are used. As frequency of occurring multiple errors are common, we need to go for advanced ECCs. Among advanced ECCs, Orthogonal Latin Squares (OLS) codes have gained renewed interest for memory protection due to their modularity and the simplicity of the decoding algorithm that enables low delay implementations. An important iss...

We define a morphism based upon a Latin square that generalizes the Thue-Morse morphism. We prove that fixed points of this morphism are overlap-free sequences, generalizing results of Allouche Shallit and Frid.

We define graph representations modulo integers and prove that any finite graph has a representation modulo some integer. We use this to obtain a new, simpler proof of Lindner, E. Mendelsohn, N. Mendelsohn, and Wolk’s result that any finite graph can be represented as an orthogonal latin square graph. Let G be a graph with vertices v,, . . . , u, and let n be a natural number. We say that G is ...

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.