Induced Subarrays of Latin Squares Without Repeated Symbols

We show that for any Latin square $L$ of order $2m$, we can partition the rows and columns of $L$ into pairs so that at most $(m+3)/2$ of the $2\times 2$ subarrays induced contain a repeated symbol. We conjecture that any Latin square of order $2m$ (where $m\geq 2$, with exactly five transposition c...

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Bibliographic Details
Published in:The Electronic journal of combinatorics 2013-03, Vol.20 (1)
Main Authors: Abel, R. Julian R., Cavenagh, Nicholas J., Kuhl, Jaromy
Format: Article
Language:English
Online Access:Get full text
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Summary:We show that for any Latin square $L$ of order $2m$, we can partition the rows and columns of $L$ into pairs so that at most $(m+3)/2$ of the $2\times 2$ subarrays induced contain a repeated symbol. We conjecture that any Latin square of order $2m$ (where $m\geq 2$, with exactly five transposition class exceptions of order $6$) has such a partition so that  every $2\times 2$ subarray induced contains no repeated symbol. We verify this conjecture by computer when $m\leq 4$.
ISSN:1077-8926
1077-8926
DOI:10.37236/2372