A diagonal Latin square is a Latin square whose main diagonal and back diagonal are both transversals. Let dr be the least integer such that for all n > dr there exist r pairwise orthogonal diagonal Latin squares of order n. In a previous paper Wallis and Zhu gave several bounds on the dr. In this paper we shall present some constructions of pairwise orthogonal diagonal Latin squares and conseq...

In this paper a certain condition on partial latin squares is shown to be sufficient to guarantee that the partial square can be completed, namely, that it have fewer than n entries, and that at most [(n + I)/21 of these lie off some line,where n is the order of the square. This is applied to establish that the Evans conjecture is true for n < 8; i.e., that given a partial latin square of order...

A critical set in a Latin square of order n is a set of entries in a Latin square which can be embedded in precisely one Latin square of order n. Also, if any element of the critical set is deleted, the remaining set can be embedded in more than one Latin square of order n. In this paper we find all the critical sets of different sizes in the Latin squares of order at most six. We count the num...

Symmetries of a partial Latin square are primarily determined by its autotopism group. Analogously to the case of Latin squares, given an isotopism Θ, the cardinality of the set PLSΘ of partial Latin squares which are invariant under Θ only depends on the conjugacy class of the latter, or, equivalently, on its cycle structure. In the current paper, the cycle structures of the set of autotopisms...

A general algorithm for finding a minimal critical set for any latin square is presented. By implementing this algorithm, minimal critical sets for all the latin squares of order six have been found. In addition, this algorithm is used to prove that the size of the minimal critical set for a back circulant latin square of order seven is twelve, and for order nine is twenty. These results provid...

A partial Latin square is premature if it has no completion, but it admits a completion after removing any of its symbols. This type of partial Latin square has been introduced by Brankovic, Honik, Miller and Rosa [Ars Combinatoria, to appear] where the authors showed that the number of empty cells in an n x n premature latin square is at least 371 4. vVe improve this lower bound to 7n/2 0(71,).

A latin trade is a subset of a latin square which may be replaced with a disjoint mate to obtain a new latin square.A d-homogeneous latin trade is one which intersects each row, each column and each entry of the latin square either 0 or d times. In this paper we give a construction for minimal d-homogeneous latin trades of size dm, for every integer d 3, and m 1.75d2 + 3. We also improve this b...

A latin transversal in a square matrix of order n is a set of entries, no two in the same row or column, which are pairwise distinct. A longstanding conjecture of Ryser states that every Latin square with odd order has a latin transversal. Some results on the existence of a large partial latin transversal can be found in [11,6,16]. Mainly motivated by Ryser’s conjecture, Erdős and Spencer [8] p...

A Greedy Defining Set is a set of entries in a Latin Square with the property that when the square is systematically filled in with a greedy algorithm, the greedy algorithm succeeds. Let g(n) be the smallest defining set for any Latin Square of order n. We give theorems on the upper bounds of gn and a table listing upper bounds of gn for small values of n. For a circulant Latin square, we find ...

It is shown that any non-solvable Latin Tableau (LT) has a solvable LT-extension with the same shape, and an algorithm computing the minimal such extension is presented. Minimal Latin Square embeddings of solvable Latin Tableaux are established. The results depend partly on the truth of the Wide Partition Conjecture for Latin Tableaux. More about Latin Tableaux and their Embeddings Bart Demoen ...