Prospects For Good Embeddings Of Pairs Of Partial Orthogonal Latin Squares And Of Partial Kirkman Triple Systems

It is known that a pair of mutually orthogonal Latin squares (MOLS) of order \(n\) can be embedded in a pair of MOLS of order \(t\) if \(t \geq 3n\). Here, we discuss the prospects of extending this result to the case when the smaller pair is only a pair of mutually orthogonal \({partial}\) Latin squares (MOPLS). We obtain some conditions, analogous to those of Ryser for embedding partial Latin squares in complete Latin squares, which we show are necessary for the embedding of MOPLS. We discuss also some implications if these conditions are, in fact, also sufficient.

We also discuss the analogous problem for pairs of partial Kirkman triple systems (PKTS).