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 \(n \geq 4\). Cavenagh and Ohman have shown that partial Latin squares of order \(4m + 1\) for \(m \geq 1\) [1] and \(4m – 1\) for \(m \geq 2\) [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.