Completing and Avoiding Partial Latin Squares.
by
Tristan Denley
University of Mississippi
Coauthors: Jaromy Kuhl, University of Western Florida

The Evans conjecture stated that partial Latin square of order n with

at most n-1 entries can be completed. Independently this was confirmed by Haggkist for n >1111, and in general by Smetanuik, and Anderson and

Hilton. We will present recent results that generalize this idea to r-multi Latin squares. In particular we shall show that a partial r-multi Latin square of order n with at most n-1 entries can be completed. Using this generalization, we confirm a case of a conjecture of Haggkvist. Similar ideas can also be used to answer questions about avoiding partial Latin squares. We shall use these ideas to show that a partial generalized sudoku square can always be avoided by some generalized sudoku. We will also mention some recent work on similar questions in higher dimensions

Date received: April 14, 2008