In this paper, we combine the notions of completing and avoiding partial latin squares. Let P be a partial latin square of order n and let Q be the set of partial latin squares of order n that avoid P . We say that P is Q -completable if P can be completed to a latin square that avoids Q ∈ Q. We prove that if P has order 4t and contains at most t − 1 entries, then P is Q -completable for each Q...

A partial difference set having parameters (n2, r(n − 1), n + r2 − 3r, r2 − r) is called a Latin square type partial difference set, while a partial difference set having parameters (n2, r(n + 1),−n + r2 + 3r, r2 + r) is called a negative Latin square type partial difference set. In this paper, we generalize well-known negative Latin square type partial difference sets derived from the theory o...

In this paper we study the concept of Latin-majorizati-\on. Geometrically this concept is different from other kinds of majorization in some aspects. Since the set of all $x$s Latin-majorized by a fixed $y$ is not convex, but, consists of union of finitely many convex sets. Next, we hint to linear preservers of Latin-majorization on $ mathbb{R}^{n}$ and ${M_{n,m}}$.

Hall’s condition is a well-known necessary condition for the existence of a proper coloring of a graph from prescribed lists. Completing a partial latin square is a very special kind of graph list-coloring problem. Cropper’s question was: is Hall’s condition sufficient for the existence of a completion of a partial latin square? The folk belief that the answer must be no is confirmed here, but,...

Suppose that n is even and a set of n 2 − 1 mutually orthogonal Latin squares of order n exists. Then we can construct a strongly regular graph with parameters (n, n 2 (n−1), n 2 ( 2 −1), n 2 ( 2 −1)), which is called a Latin square graph. In this paper, we give a sufficient condition of the Latin square graph for the existence of a projective plane of order n. For the existence of a Latin squa...

Let n and k be integers, with n > 1 and k > 0. An (n×n)/k semiLatin square S is an n × n array, whose entries are k-subsets of an nk-set, the set of symbols of S, such that each symbol of S is in exactly one entry in each row and exactly one entry in each column of S. SemiLatin squares form an interesting class of combinatorial objects which are useful in the design of comparative experiments. ...

We define the notions of nest and influence of a subset of a critical set of a back circulant latin square, and study their properties. We also show that a secret sharing scheme based on a critical set of a latin square is both compartmentalised, and hierarchical.

Sudoku has risen in popularity over the past few years. The rules are simple, yet the solutions are often less than trivial. Mathematically, these puzzles are interesting in their own right. This paper will generalize the idea of a sudoku puzzle to define a new kind of n× n array. We define a latin square of order n as an n×n array where every row and every column contain every symbol 1, 2, . ....

A (partial) Latin square is a table of multiplication of a (partial) quasigroup. Multiplication of a (partial) quasigroup may be considered as a set of triples. We give a necessary and sufficient condition when a set of triples is a quotient of a (partial) Latin square.

We present new abelian partial difference sets and amorphic group schemes of both Latin square type and negative Latin square type in certain abelian p-groups. Our method is to construct what we call pseudo-quadratic bent functions and use them in place of quadratic forms. We also discuss a connection between strongly regular bent functions and amorphic group schemes.