A Census of Critical Sets in the Latin Squares of Order at Most Six

Abstract

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 \(6 \times 6\) case. Finally some examples are given of each type of critical set.