Induced Subarrays of Latin Squares Without Repeated Symbols

Abstract

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 $m\leq 4$.