We consider the problem of constructing Latin cubes subject to the condition that some symbols may not appear in certain cells. We prove that there is a constant $\gamma > 0$ such that if $n=2^t$ and $A$ is a $3$-dimensional $n\times n\times n$ array where every cell contains at most $\gamma n$ symbols, and every symbol occurs at most $\gamma n$ times in every line of $A$, then $A$ is avoidable; that is, there is a Latin cube $L$ of order $n$ such that for every $1\leq i,j,k\leq n$, the symbol in position $(i,j,k)$ of $L$ does not appear in the corresponding cell of $A$.
Carl Johan Casselgren
1
;
Klas Markström
2
;
Lan Anh Pham
2
1
Dept. of Mathematics, Linköping University
2
Umeå University
Carl Johan Casselgren; Klas Markström; Lan Anh Pham. Latin cubes with forbidden entries. The electronic journal of combinatorics, Tome 26 (2019) no. 1. doi: 10.37236/8157
@article{10_37236_8157,
author = {Carl Johan Casselgren and Klas Markstr\"om and Lan Anh Pham},
title = {Latin cubes with forbidden entries},
journal = {The electronic journal of combinatorics},
year = {2019},
volume = {26},
number = {1},
doi = {10.37236/8157},
zbl = {1409.05038},
url = {http://geodesic.mathdoc.fr/articles/10.37236/8157/}
}
TY - JOUR
AU - Carl Johan Casselgren
AU - Klas Markström
AU - Lan Anh Pham
TI - Latin cubes with forbidden entries
JO - The electronic journal of combinatorics
PY - 2019
VL - 26
IS - 1
UR - http://geodesic.mathdoc.fr/articles/10.37236/8157/
DO - 10.37236/8157
ID - 10_37236_8157
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%A Lan Anh Pham
%T Latin cubes with forbidden entries
%J The electronic journal of combinatorics
%D 2019
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%N 1
%U http://geodesic.mathdoc.fr/articles/10.37236/8157/
%R 10.37236/8157
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