Product Constructions For Critical Sets In Latin Squares

Abstract

Let $T$ be a partial Latin square. If there exist two distinct Latin squares $M$ and $N$ of the same order such that $M\cap N=T$, then $M\setminus T$ is said to be a Latin trade. For a given Latin square $M$, it is possible to identify a subset of entries, termed a critical set, which intersects all Latin trades in $M$ and is minimal with respect to this property.

Stinson and van Rees have shown that under certain circumstances, critical sets in Latin squares $M$ and $N$ can be used to identify critical sets in the direct product $M\times N$. This paper presents a refinement of Stinson and van Rees’ results and applies this theory to prove the existence of two new families of critical sets.