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 \(\{1, 2, \ldots, n\}\) 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.