Avoiding \((m,m,m)\)-arrays of order \(n=2^k\)
The electronic journal of combinatorics, Tome 19 (2012) no. 1
An $(m,m,m)$-array of order $n$ is an $n\times n$ array such that each cell is assigned a set of at most $m$ symbols from $\left\{1,\dots ,n\right\}$ such that no symbol occurs more than $m$ times in any row or column. An $ (m,m,m)$-array is called avoidable if there exists a Latin square such that no cell in the Latin square contains a symbol that also belongs to the set assigned to the corresponding cell in the array. We show that there is a constant $\gamma $ such that if $m\le\gamma 2^k$ and $k\ge14$, then any $(m,m,m)$-array of order $n=2^k$ is avoidable. Such a constant $\gamma$ has been conjectured to exist for all $n$ by Häggkvist.
@article{10_37236_2167,
author = {Lina J. Andr\'en},
title = {Avoiding \((m,m,m)\)-arrays of order \(n=2^k\)},
journal = {The electronic journal of combinatorics},
year = {2012},
volume = {19},
number = {1},
doi = {10.37236/2167},
zbl = {1243.05047},
url = {http://geodesic.mathdoc.fr/articles/10.37236/2167/}
}
Lina J. Andrén. Avoiding \((m,m,m)\)-arrays of order \(n=2^k\). The electronic journal of combinatorics, Tome 19 (2012) no. 1. doi: 10.37236/2167
Cité par Sources :