Bachelor latin squares with large indivisible plexes

Abstract

In a latin square of order n, a k-plex is a selection of kn entries in which each row, column, and symbol occurs k times. A 1-plex is also called a transversal. A k-plex is indivisible if it contains no c-plex for 0<c<k. We prove that, for all n≥4, there exists a latin square of order nthat can be partitioned into an indivisible ⌊n/2⌋-plex and a disjoint indivisible ⌈n/2⌉-plex. For all n≥3, we prove that there exists a latin square of order n with two disjoint indivisible ⌊n/2⌋-plexes. We also give a short new proof that, for all odd n≥5, there exists a latin square of order n with at least one entry not in any transversal. Such latin squares have no orthogonal mate. Copyright © 2011 Wiley Periodicals, Inc. J Combin Designs 19:304-312, 2011