Bachelor latin squares with large indivisible plexes

✍ Scribed by Judith Egan

• Publisher: John Wiley and Sons
• Year: 2011
• Tongue: English
• Weight: 116 KB
• Volume: 19
• Category: Article
• ISSN: 1063-8539
• DOI: 10.1002/jcd.20281

✦ Synopsis

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 n that 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.