Critical sets for families of Latin squares

In this paper I give details of new constructions for critical sets in latin squares. These latin squares, of order n, are such that they can be partitioned into four subsquares each of which is based on the addition table of the integers module n/2, an isotopism of this or a conjugate.

Critical sets in direct products of back circulant Latin squares

To date very Few families of critical sets for latin squares are known. The only previously known method for constructing critical sets involves taking a critical set which is known to satisfy certain strong initial conditions and using a doubling construction. This construction can be applied to the known critical sets in back circulant latin squares of even order. However, the doubling construction cannot be applied to critical sets in back circulant latin squares of odd order. In this paper a family of critical sets is identified for latin squares which are the product of the latin square of order 2 with a back circulant latin square of odd order. The proof that each element of the critical set is an essential part of the reconstruction process relies on the proof of the existence of a large number of latin interchanges.

Smallest weak and smallest totally weak critical sets in the latin squares of order at most seven

A critical set in a latin square of order n is a set of entries in a latin square which can be embedded in precisely one latin square of order n. Also, if any element of the critical set is deleted, the remaining set can be embedded in more than one latin square of order n. In this paper we find smallest weak and smallest totally weak critical sets for all the latin squares of orders six and seven. Moreover, we computationally prove that there is no (totally) weak critical set in the back circulant latin square of order five and we find a totally weak critical set of size seven in the other main class of latin squares of order five.

Balanced critical sets in Latin squares

In this note we first introduce balanced critical sets and near balanced critical sets in Latin squares. Then we prove that there exist balanced critical sets in the back circulant Latin squares of order 3n for n even. Using this result we decompose the back circulant Latin squares of order 3n, n even, into three isotopic and disjoint balanced critical sets each of size 3n. We also find near balanced critical sets in the back circulant Latin squares of order 3n for n odd. Finally, we examine representatives of each main class of Latin squares of order up to six in order to determine which main classes contain balanced or near balanced critical sets.

A census of critical sets in the Latin squares of order at most six

A critical set in a Latin square of order n is a set of entries from the square which can be embedded in precisely one Latin square of order n, Such that if any element of the critical set. is deleted, the remaining set can be embedded, in more than one Latin square of order n.. In this paper we find all the critical sets of different sizes in the Latin squares of order at most six. We count the number of main and isotopy classes of these critical sets and classify critical sets from the main classes into various strengths. Some observations are made about the relationship between the numbers of classes, particularly in the 6 x 6 case. Finally some examples are given of each type of critical set.

An investigation of 2-critical sets in latin squares

In this paper we focus on the existence of 2-critical sets in the latin square corresponding to the elementary abelian 2-group of order 2(n). It has been shown by Stinson and van Rees that this latin square contains a 2-critical set of volume 4(n) - 3(n). We provide constructions for 2-critical sets containing 4(n) - 3(n) + 1 - (2(k-1) + 2(m-1) + 2(n-(k+m+1))) entries, where 1 less than or equal to k less than or equal to n and 1 less than or equal to m less than or equal to n - k. That is, we construct 2-critical sets for certain values less than 4(n) - 3(n) + 1 - 3 (.) 2([n /3]-1). The results raise the interesting question of whether, for the given latin square, it is possible to construct 2-critical sets of volume m, where 4(n) - 3(n) + 1 - 3 (.) 2([n/3]-1) < m < 4(n) - 3(n).

A new class of critical sets in Latin squares

Previously the process of finding critical sets in Latin squares has been inside cumbersome by the complexity and number of Latin trades that, must be constructed. In this paper we develop a theory of Latin trades that yields more transparent constructions. We use these Latin trades to find a new class of critical sets for Latin squares which are a product of the Latin square of order 2 with a. back circulant Latin square of odd order.