Atomic latin squares have indivisible structure which mimics that of the cyclic groups of prime order. They are related to perfect 1-factorisations of complete bipartite graphs. Only one example of an atomic latin square of a composite order (namely 27) was previously known. We show that this one example can be generated by an established method of constructing latin squares using cyclotomic or...

Let n be an integer. We show the existence of at least three non-isomorphic non-commutative Latin squares of order n which are embeddable in groups when n ≥ 5 is odd. By using a similar construction for the case when n ≥ 4 is even, we show that certain non-commutative Latin squares of order n are not embeddable in groups. Keywords—group, Latin square, embedding.

A Latin square L(n, k) is a square of order n with its entries colored with k colors so that all the entries in a row or column have different colors. Let d(L(n, k)) be the minimal number of colored entries of an n × n square such that there is a unique way of coloring of the yet uncolored entries in order to obtain a Latin square L(n, k). In this paper we discuss the properties of d(L(n, k)) f...

Let G be a finite group of order n ≥ 2 and T be the Cayley table of G. Obviously T is a Latin square. In this paper we study submatrices of T which are Latin squares. Let Lm(G) where m ≤ n be the number of Latin squares of order m in T . We compute Lm(G) and study some properties of such submatrices. Also we classify the Latin squares of order m in T and compute the number of each classes in so...

Given a Latin square L and a subset F of its autotopism group U(L), we study in this paper some properties and results which partial Latin squares contained in L inherit from U(L), by using F. In particular, we define the concept of F-critical set of L and we ask ourselves about the smallest one contained in L.

A k-plex of a latin square is a collection of cells representing each row, column, and symbol precisely k times. The classic case of k = 1 is more commonly known as a transversal. We introduce the concept of a k-weight, an integral weight function on the cells of a latin square whose row, column, and symbol sums are all k. We then show that several non-existence results about k-plexes can been ...

Forty years ago, Goethals and Seidel showed that if the adjacency algebra of a strongly regular graph X contains a Hadamard matrix then X is of Latin square type or of negative Latin square type [8]. We extend their result to complex Hadamard matrices and find only three additional families of parameters for which the strongly regular graphs have complex Hadamard matrices in their adjacency alg...

A Latin square L = L(`ij) 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) = `ij for all i, j ∈ Zn; 2. there are no polynomials Vj(x) ∈ Zn[x] such that Vj(i) = `ij for all i, j ∈ Zn. In the presented paper we describe four possible constructions of such Latin squares which might be of particular interes...

Al2O3 is commonly used as host material for Er-doped integrated optical amplifiers. In this paper, a Graeco-Latin square is used in DC reactive magnetron sputtering deposition experiment in order to get low optical losses Al2O3 layer. By reasonable selection and careful arrangement of experimental parameters, an optimal combination of deposition parameters is obtained via statistic analysis wit...