Extending Partial Latin Cubes

Abstract

In the spirit of Ryser’s theorem, we prove sufficient conditions on \(k\), \(\ell\), and \(m\) so that \(k \times \ell \times m\) Latin boxes, i.e., partial Latin cubes whose filled cells form a \(k \times \ell \times m\) rectangular box, can be extended to a \(k \times n \times m\) Latin box, and also to a \(k \times n \times m\) Latin box, where \(n\) is the number of symbols used, and likewise the order of the Latin cube. We also prove a partial Evans-type result for Latin cubes, namely that any partial Latin cube of order \(n\) with at most \(n-1\) filled cells is completable, given certain conditions on the spatial distribution of the filled cells.