Uniform critical sets in Latin squares

Abstract

In this paper, we introduce two new classes of critical sets, $t$-uniform and $T$-uniform (where $t$ is a positive integer and $T$ is a partial Latin square). We identify, up to isomorphism, all $t$-uniform critical sets of order $n$, where $2\le n\le 6$. We show that the completable product of two $T$-uniform critical sets is a $T$-uniform critical set for certain partial Latin squares $T$, and then apply this theorem to small examples to generate infinite families of $T$-uniform critical sets.