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Abstract
A kplex in a latin square of order n is a selection of kn entries that includes k representatives from each row and column and k occurrences of each symbol. A 1plex is also known as a transversal.
It is well known that if n is even then B_{n}, the addition table for the integers modulo n, possesses no transversals. We show that there are a great many latin squares that are similar to B_{n} and have no transversal. As a consequence, the number of species of transversalfree latin squares is shown to be at least n^{n3/2 (1/2−o(1))} for even n → ∞.
We also produce various constructions for latin squares that have no transversal but do have a kplex for some odd k > 1. We prove a 2002 conjecture of the second author that for all even orders n > 4 there is a latin square of order n that contains a 3plex but no transversal. We also show that for odd k and m ≥ 2, there exists a latin square of order 2km with a kplex but no k′plex for odd k′ < k.
Original language  English 

Article number  #P2.45 
Number of pages  15 
Journal  Electronic Journal of Combinatorics 
Volume  24 
Issue number  2 
Publication status  Published  30 Jun 2017 
Keywords
 Latin square
 Plex
 Transversal
 Triplex
Projects
 1 Finished

Matchings in Combinatorial Structures
Wanless, I., Bryant, D. & Horsley, D.
Australian Research Council (ARC), Monash University, University of Queensland , University of Melbourne
1/01/15 → 10/10/20
Project: Research