On The Spectrum of Critical Sets in Back Circulant Latin Squares

Abstract

In this paper we prove that there exists a strong critical set of size \(m\) in the back circulant latin square of order \(n\) for all \(\frac{n^2-1}{2} \leq m \leq \frac{n^2-n}{2}\), when \(n\) is odd. Moreover, when \(n\) is even we prove that there exists a strong critical set of size \(m\) in the back circulant latin square of order \(n\) for all \(\frac{n^2-n}{2}-(n-2) \leq m \leq \frac{n^2-n}{2}\) and \(m \in \{\frac{n^2}{4}, \frac{n^2}{4}+2, \frac{n^2}{4}+4, \ldots, \frac{n^2-n}{2}-n\}\).