We investigate defining sets for latin squares where we are given that the latin square is the Cayley table for some group. Our main result is that the proportion of entries in a smallest defining set approaches zero as the order of the group increases without bound.

In this paper we prove that a finite partial commutative (idempotent commutative) Latin square can be embedded in a finite commutative (idempotent commutative) Latin square. These results are then used to show that the loop varieties defined by any non-empty subset of the identities {x(xy) = y, (yx)x = y} and the quasi-group varieties defined by any non-empty subset of {x” = x, x(xy) = y, (yx)x...

We show that for any Latin square L of order 2m, we can partition the rows and columns of L into pairs so that at most (m + 3)/2 of the 2 × 2 subarrays induced contain a repeated symbol. We conjecture that any Latin square of order 2m (where m > 2, with exactly five transposition class exceptions of order 6) has such a partition so that every 2× 2 subarray induced contains no repeated symbol. W...

We introduce the notion of premature partial latin squares; these cannot be completed, but if any of the entries is deleted, a completion is possible. We study their spectrum, i.e., the set of integers t such that there exists a premature partial latin square of order n with exactly t nonempty cells.

We identify a weak critical set in each cyclic latin square of order greater than 5. This provides the first example of an infinite family of weak critical sets. The proof uses several constructions for latin interchanges which are generalisations of those introduced by Donovan and Cooper.

In parallel/distributed computing systems, the all-to-all personalized communication (or complete exchange) is required in numerous applications of parallel processing. In this paper, we consider this problem for logN stage Multistage Interconnection Networks (MINs). It is proved that the set of admissible permutations for a MIN can be partitioned in Latin Squares. Since routing permutations be...

Author: Jenny Zhang First, let’s preview what mutually orthogonal Latin squares are. Two Latin squares L1 = [aij ] and L2 = [bij ] on symbols {1, 2, ...n}, are said to be orthogonal if every ordered pair of symbols occurs exactly once among the n2 pairs (aij , bij), 1 ≤ i ≤ n, 1 ≤ j ≤ n. Now, let me introduce a related concept which is called transversal. A transversal of a Latin square is a se...

An n× n partial Latin square P is called α-dense if each row and column has at most αn non-empty cells and each symbol occurs at most αn times in P . An n× n array A where each cell contains a subset of {1, . . . , n} is a (βn, βn, βn)-array if each symbol occurs at most βn times in each row and column and each cell contains a set of size at most βn. Combining the notions of completing partial ...