Signs on Latin Squares

Square-root cancellation for the signs of Latin squares

On Completing Latin Squares

We present a ( 2 3 − o(1))-approximation algorithm for the partial latin square extension (PLSE) problem. This improves the current best bound of 1− 1 e due to Gomes, Regis, and Shmoys [5]. We also show that PLSE is APX-hard. We then consider two new and natural variants of PLSE. In the first, there is an added restriction that at most k colors are to be used in the extension; for this problem,...

On Orthogonality of Latin Squares

Abstract: A Latin square arrangement is an arrangement of s symbols in s rows and s columns, such that every symbol occurs once in each row and each column. When two Latin squares of same order superimposed on one another, then in the resultant array every ordered pair of symbols occurs exactly once, then the two Latin squares are said to be orthogonal. A frequency square M of type F (n; λ) is ...

On Biembeddings of Latin Squares

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 Abelian 2-groups Ck 2 (k 6= 2). In turn, these biembeddings enable us to increase the best known ...

On Non-Polynomial Latin Squares

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...