A critical set in a latin square is a set of entries in a latin square which can be embedded in only one latin square. Also, if any element of the critical set is deleted, the remaining set can be embedded in more than one latin square. A critical set is strong if the embedding latin square is particularly easy to find because the remaining squares of the latin square are " forced " one at a ti...

The problem of completing partial latin squares to latin squares of the same order has been studied for many years. For instance, in 1960 Evans [9] conjectured that every partial latin square of order n containing at most n− 1 filled cells is completable to a latin square of order n. This conjecture was shown to be true by Lindner [12] and Smetaniuk [13]. Recently, Bryant and Rodger [6] establi...

A k × n Latin rectangle is a k × n array of numbers such that (i) each row is a permutation of [n] = {1, 2, . . . , n} and (ii) each column contains distinct entries. If the first row is 12 · · ·n, the Latin rectangle is said to be reduced. Since the number k × n Latin rectangles is clearly n! times the number of reduced k× n Latin rectangles, we shall henceforth consider only reduced Latin rec...

The chromatic number of a Latin square is the least number of partial transversals which cover its cells. This is just the chromatic number of its associated Latin square graph. Although Latin square graphs have been widely studied as strongly regular graphs, their chromatic numbers appear to be unexplored. We determine the chromatic number of a circulant Latin square, and find bounds for some ...

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...

We generalize the notion of orthogonal latin squares to colorings of simple graphs. Two n-colorings of a graph are said to be orthogonal if whenever two vertices share a color in one coloring they have distinct colors in the other coloring. We show that the usual bounds on the maximum size of a certain set of orthogonal latin structures such as latin squares, row latin squares, equi-n squares, ...

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...

Suppose that L is a latin square of order m and P ⊆ L is a partial latin square. If L is the only latin square of order m which contains P , and no proper subset of P has this property, then P is a critical set of L. The critical set spectrum problem is to determine, for a given m, the set of integers t for which there exists a latin square of order m with a critical set of size t. We outline a...

Cycle switches are the simplest changes which can be used to alter latin squares, and as such have found many applications in the generation of latin squares. They also provide the simplest examples of latin interchanges or trades in latin square designs. In this paper we construct graphs in which the vertices are classes of latin squares. Edges arise from switching cycles to move from one clas...