Maximal partial Latin cubes (Q2341074)

Summary: 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.