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The completion of partial Latin squares

In this thesis, the problem of completing partial latin squares is examined. In particular, the completion problem for three primary classes of partial latin squares is investigated. First, the theorem of Marshall Hall regarding completions of latin rectangles is discussed. Secondly, a proof of Evans’ conjecture is presented, which deals with partial latin squares of order n containing at most n − 1 entries. Finally, we investigate an open problem regarding completions of k-staggers, which are partial latin squares in which each row and column contains exactly k entries, and each value is used exactly k times. For this final problem, results are presented for a number of special cases. Computational results are obtained, and on the basis of these, conjectures are drawn. Finally, possible methods of proof for these conjectures are discussed.