Maximal Partial Latin Cubes

Abstract

We prove that each maximal partial Latin cube must have more than $29.289\%$ of its cells filled and show by construction that this is a nearly tight bound. We also prove upper and lower bounds on the number of cells containing a fixed symbol in maximal partial Latin cubes and hypercubes, and we use these bounds to determine for small orders $n$ the numbers $k$ for which there exists a maximal partial Latin cube of order $n$ with exactly $k$ entries. Finally, we prove that maximal partial Latin cubes of order $n$ exist of each size from approximately half-full ($n^3/2$ for even $n\geq 10$ and $(n^3+n)/2$ for odd $n\geq 21$) to completely full, except for when either precisely $1$ or $2$ cells are empty.