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.
@article{10_37236_4726,
author = {Thomas Britz and Nicholas J. Cavenagh and Henrik Kragh S{\o}rensen},
title = {Maximal partial {Latin} cubes},
journal = {The electronic journal of combinatorics},
year = {2015},
volume = {22},
number = {1},
doi = {10.37236/4726},
zbl = {1310.05032},
url = {http://geodesic.mathdoc.fr/articles/10.37236/4726/}
}
TY - JOUR
AU - Thomas Britz
AU - Nicholas J. Cavenagh
AU - Henrik Kragh Sørensen
TI - Maximal partial Latin cubes
JO - The electronic journal of combinatorics
PY - 2015
VL - 22
IS - 1
UR - http://geodesic.mathdoc.fr/articles/10.37236/4726/
DO - 10.37236/4726
ID - 10_37236_4726
ER -
%0 Journal Article
%A Thomas Britz
%A Nicholas J. Cavenagh
%A Henrik Kragh Sørensen
%T Maximal partial Latin cubes
%J The electronic journal of combinatorics
%D 2015
%V 22
%N 1
%U http://geodesic.mathdoc.fr/articles/10.37236/4726/
%R 10.37236/4726
%F 10_37236_4726
Thomas Britz; Nicholas J. Cavenagh; Henrik Kragh Sørensen. Maximal partial Latin cubes. The electronic journal of combinatorics, Tome 22 (2015) no. 1. doi: 10.37236/4726
