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Bachelor latin squares with large indivisible plexes
Author(s) -
Egan Judith
Publication year - 2011
Publication title -
journal of combinatorial designs
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.618
H-Index - 34
eISSN - 1520-6610
pISSN - 1063-8539
DOI - 10.1002/jcd.20281
Subject(s) - latin square , disjoint sets , transversal (combinatorics) , combinatorics , mathematics , order (exchange) , square (algebra) , bachelor , selection (genetic algorithm) , computer science , law , chemistry , artificial intelligence , political science , mathematical analysis , rumen , geometry , food science , finance , fermentation , economics
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. Copyright © 2011 Wiley Periodicals, Inc. J Combin Designs 19:304‐312, 2011