A theory about Latin Squares following [1]. A Latin Square is a n×n table filled with integers from 1 to n where each number appears exactly once in each row and each column. A Latin Rectangle is a partially filled n × n table with r filled rows and n − r empty rows, such that each number appears

In recent times there has been some interest in studying partial latin squares which have no completions or precisely one completion, and which are critical with respect to this property. Such squares are called, respectively, premature partial latin squares and critical sets. There has also been

A complete classification of redundant sets of inequalities in the specification of the Latin Square problem of size N is proven. Related issues on variations of the same problem are discussed. 1

A Latin square L = L(ℓij) over the set S = {0, 1,..., n − 1} is called totally non-polynomial over Zn iff 1. there are no polynomials Ui(y) ∈ Zn[y] such that Ui(j) = ℓij for all i, j ∈ Zn; 2. there are no polynomials Vj(x) ∈ Zn[x] such that Vj(i) = ℓij for all i, j ∈ Zn. In the presented paper

The logarithm of the maximum number of transversals over all latin squares of order n is greater than n6 (lnn+O(1)).

We (1) determine the number of Latin rectangles with 11 columns and each possible number of rows, including the Latin squares of order 11, (2) answer some questions of Alter by showing that the number of reduced Latin squares of order n is divisible by f! where f is a particular integer close to 1

A square array is avoidable if for each set of n symbols there is an n × n Latin square on these symbols which differs from the array in every cell. The main result of this paper is that for m ≥ 2 any partial Latin square of order 4m − 1 is avoidable, thus concluding the proof that any partial

A finite latin square is an n × n matrix whose entries are elements of the set {1,...,n} and no element is repeated in any row or column. Given equivalence relations on the set of rows, the set of columns, and the set of symbols, respectively, we can use these relations to identify equivalent rows

A known construction for face 2-colourable triangular embeddings of complete regular tripartite graphs is re-examined from the viewpoint of the underlying Latin squares. This facilitates biembeddings of a wide variety of Latin squares, including those formed from the Cayley tables of the elementary

A latin square of order n is an n×n array of n symbols in which each symbol occurs exactly once in each row and column. A transversal of such a square is a set of n entries such that no two entries share the same row, column or symbol. Transversals are closely related to the notions of complete