Avoidable Partial Latin Squares of Order $4m+1$

Abstract

A partial Latin square $P$ of order $n$ is an $n\times n$ array with entries from the set $\{1,2,\dots ,n\}$ such that each symbol is used at most once in each row and at most once in each column. If every cell of the array is filled, we call $P$ a Latin square. A partial Latin square $P$ of order $n$ is said to be avoidable if there exists a Latin square $L$ of order $n$ such that $P$ and $L$ are disjoint. That is, corresponding cells of $P$ and $L$ contain different entries. In this note, we show that, with the trivial exception of the Latin square of order $1$, every partial Latin square of order congruent to $1$ modulo $4$ is avoidable.