On the completability of incomplete Latin squares (Q966163)

An \textit{incomplete Latin square} is an \(n \times n\) array where some of the cells have an entry from \(\{1,\dots,n\}\) such that no symbol occurs twice in a row or column. The square is \textit{completable} if it can be extended to a Latin square of the same order. The question is {`Which incomplete Latin squares are completable?'} In this paper the authors examine when an incomplete Latin square has a row that can be completed. They introduce an availability matrix and use the Frobenius-König Theorem to find necessary and sufficient conditions for such a row. They then examine when the incomplete square has two rows that can be extended using the framework of \((1,2)\)-permutations. They consider an integer programming formulation with polyhedral results, and a nice application for class-teacher time-table problems.