Smallest Critical Sets for the Latin Squares of Orders Six and 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 \(n\). A smallest critical set in a Latin square is a critical set of minimum cardinality. In this paper we find smallest critical sets for all the Latin squares of orders six and seven. We also find smallest critical sets of orders six and seven which are also weak critical sets. In particular, we find a weak critical set of size twelve for the dihedral group of order six.