A few remarks on avoiding partial Latin squares

Ars Combinatoria

0381-7032

106

313

316

2007

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Let P be an n x n array of symbols. P is called avoidable if for every set of n symbols, there is an n x n Latin square Lon these symbols so that corresponding cells in Land P differ. Due to recent work of Cavenagh and Olunan, we now know that all n x n partial Latin squares are avoidable for n 2: 4. Cavenagh and Ohman have shown that partial Latin squares of order 4m + 1 form 2: 1 [lJ and 4m -1 form 2: 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 x n partial Latin square P with some specified structure, is there an n X n Latin square L of the same structure for which L avoids P? We answer this question in the context of generalized sudoku squares.

Latin squares

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