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.