A Few Remarks on Avoiding Partial Latin Squares

Abstract

Let $P$ be an $n\times n$ array of symbols. $P$ is called avoidable if for every set of $z$ symbols, there is an $n\times n$ Latin square $L$ on these symbols so that corresponding cells in $P$ and $L$ differ. Due to recent work of Cavenagh and Ohman, we now know that all $n\times n$ partial Latin squares are avoidable for $n\ge 4$. Cavenagh and Ohman have shown that partial Latin squares of order $4m+1$ for $m\ge 1$ [1] and $4m–1$ for $m\ge 2$ [2] are avoidable. We give a short argument that includes all partial Latin squares of these orders of at least $9$. We then ask the following question: given an $n\times n$ partial Latin square $P$ with some specified structure, is there an $n\times n$ Latin square $L$ of the same structure for which $L$ avoids $P$? We answer this question in the context of generalized sudoku squares.