Indivisible partitions of 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 occurs k times. A 1-plex is also called a transversal. An indivisible k-plex is one that contains no c-plex for 0 < c < k. For orders n not in {2,6}, existence of latin squares with a decomposition into 1-plexes was famously shown in 1960 by Bose, Shrikhande and Parker. A main result of this paper is that if k is a proper divisor of n then there exists a latin square of order n composed of disjoint indivisible k-plexes.

Define κ(n) to be the largest integer k such that some latin square of order n contains an indivisible k-plex. We report on extensive computations of indivisible plexes and partitions in latin squares of order at most 9. We determine κ(n) exactly for n ≤ 8 and find that κ(9) in {6,7}. In the process we confirm that the latin squares of order 9 satisfy a conjecture by Rodney that every latin square contains a 2-plex.

For each group table of order n ≤ 8 we report the number of indivisible plexes and indivisible partitions. For group tables of order 9 we give the number of indivisible plexes and identify which types of indivisible partitions occur.

By extending an argument used by Mann, we show that for all n ≥ 5 there is some k in {1,2,3,4} for which there exists a latin square of order n that has k disjoint transversals and a disjoint (n-k)-plex that contains no c-plex for any odd c.

Click here to download the whole paper.