Indivisible plexes in latin squares

Abstract

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.