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A generalisation of transversals for Latin squares

We define a k-plex to be a partial latin square of order n
containing kn entries such that exactly k entries lie in each row
and column and each of n symbols occurs exactly k times. A
transversal of a latin square corresponds to the case k=1. For
k>n/4 we prove that not all k-plexes are completable to latin
squares. Certain latin squares, including the Cayley tables of many
groups, are shown to contain no (2c+1)-plex for any integer c.
However, Cayley tables of soluble groups have a 2c-plex for each
possible c. We conjecture that this is true for all latin squares
and confirm this for orders n≤8. Finally, we demonstrate the
existence of indivisible k-plexes, meaning that they contain
no c-plex for 1≤c<k.
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