In this paper we study systems of distinct representatives (SDR’s) and Latin squares, considering SDR’s especially in their application to constructing Latin squares. We give proofs of several important elementary results for SDR’s and Latin squares, in particular Hall’s marriage theorem and lower bounds for the number of Latin squares of each order, and state several other results, such as nec...

IsotoIHsl:n that lower bounds for infinite N of order n. hand or computer, strong critical sets of minimum nOn-lS01110Tnitnc reduced latin squares of order less than or of order 6. These latin squares have been taken from 129 onwards. We summarise our results in Table the numbering system listed by Denes and Keedwell latin squares. This numbering system is based on the ,<:n,,-r1i',rrHl1<:rn cla...

A double latin square of order 2n on symbols s1;y; sn is a 2n 2n matrix A 1⁄4 ðaijÞ in which each aij is one of the symbols s1;y; sn and each sk occurs twice in each row and twice in each column. For k 1⁄4 1;y; n let BðA; skÞ be the bipartite graph with vertices r1;y; r2n; c1;y; c2n and 4n edges 1⁄2ri; cj corresponding to ordered pairs ði; jÞ such that aij 1⁄4 sk: We say that A is Hamiltonian i...

Let n and k be integers, with n > 1 and k > 0. An (n × n)/k semi-Latin square S is an n × n array, whose entries (called blocks) are k-element subsets of a set of size nk, the set of symbols of S, such that each symbol of S occurs exactly once in each row and exactly once in each column of S. Semi-Latin squares form a class of designs generalising Latin squares, and have applications in areas i...

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 interest in related maximal partial latin squares. This paper will explore the connection between t...

In the pandiagonal Latin Square problem, a square grid of size N needs to be filled with N types of objects, so that each column, row, and wrapped around diagonal (both up and down) contains an object of each type. This problem dates back to at least Euler. In its specification as a constraint satisfaction problem, one uses the all different constraint. The known redundancy result about all dif...

A Room t-cube is standardized if it also satisfies (iv) the ith diagonal cell contains the pair of symbols {co, i}. There are several other equivalent forms that are taken by Room t-cubes, one in terms of graphs and one in terms of Latin squares. The first of these forms is as a set of orthogonal one-factorizations of K,,. Let G be a graph with an even number of vertices. A one-factor in G is a...