two latin squares of order $n$ are orthogonal if in their superposition, each of the$n^{2}$ ordered pairs of symbols occurs exactly once. colbourn, zhang and zhu, in a seriesof papers, determined the integers $r$ for which there exist a pair of latin squares oforder $n$ having exactly $r$ different ordered pairs in their superposition. dukes andhowell defined the same problem for latin squares ...

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 ${m_{n,m}}$.

A multi-latin square of order n and index k is an n×n array of multisets, each of cardinality k, such that each symbol from a fixed set of size n occurs k times in each row and k times in each column. A multi-latin square of index k is also referred to as a k-latin square. A 1-latin square is equivalent to a latin square, so a multi-latin square can be thought of as a generalization of a latin ...

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...

In this paper we introduce new models of random graphs, arising from Latin squares which include random Cayley graphs as a special case. We investigate some properties of these graphs including their clique, independence and chromatic numbers, their expansion properties as well as their connectivity and Hamiltonicity. The results obtained are compared with other models of random graphs and seve...

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. Nearly all known constructions of negative Latin square partial difference sets are in elementary abelian groups. In this pape...

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...

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...