Indivisible plexes in latin squares

A k-plex is a selection of kn entries of a latin square of order n in which each row, column and symbol is represented precisely k times. A transversal of a latin square corresponds to the case k=1. A k-plex is said to be indivisible if it contains no c-plex for any 0 < c < k. We prove that if n=2km for integers k ≥ 2 and m ≥ 1 then there exists a latin square of order n composed of 2m disjoint indivisible k-plexes. Also, for positive integers k and n satisfying n=3k, n=4k or n ≥ 5k, we construct a latin square of order n containing an indivisible k-plex.

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