DMS Graduate Student Seminar

Speaker:  Stacie Baumann 

Title: Completing partial latin squares with prescribed diagonal

Abstract: Necessary and sufficient numerical conditions are known for the embedding of an incomplete latin square \(L\) of order \(n\) into a latin square \( T\) of order \(t ≥ 2n+1\) in which each symbol is prescribed to occur in a given number of cells on the diagonal, \(D\), of \(T\) outside of \(L\). This includes the classic case where \(T\) is required to be idempotent.

If  \(t < 2n\), then no such numerical sufficient conditions exist since it is known that the arrangement of symbols within the given incomplete latin square can determine the embeddibility. All examples where the arrangement is a factor share the common feature that one symbol is prescribed to appear exactly once in \(D\), resulting in a conjecture over 30 years ago stating that it is only this feature that prevents numerical conditions sufficing for all \(t ≥ n\).

We prove this conjecture, providing necessary and sufficient numerical conditions for the embedding of an incomplete latin square \(L\) of order n into a latin square \(T\) of order \(t\) for all \(t ≥ n\) in which \(D\) is prescribed in the case where no symbol is required to appear exactly once in \(D\).