Smallest Weak and Smallest Totally Weak Critical Sets in the Latin Squares of Order at Most Seven

Abstract

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.