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      Latin squares with no transversals

      Cavenagh, Nicholas J.; Wanless, Ian M.
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      Cavenagh, N. J., & Wanless, I. M. (2017). Latin squares with no transversals. Electronic Journal of Combinatorics, 24(2).
      Permanent Research Commons link: https://hdl.handle.net/10289/11995
      Abstract
      Ak-plex in a latin square of ordernis a selection of kn entries that includes k representatives from each row and column and k occurrences of each symbol. A 1-plex is also known as a transversal.

      It is well known that if n is even then Bₙ, 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 Bₙ and have no transversal. As a consequence, the number of species of transversal-free latin squares is shown to be at least nⁿ³⁼²⁽¹⁼²ᵒ⁽¹⁾ for even n→∞.

      We also produce various constructions for latin squares that have no transversal but do have a k-plex 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 3-plex but no transversal. We also show that for odd k and m≥2, there exists a latin square of order 2km with a k-plex but no k'-plex for odd k'< k.
      Date
      2017
      Type
      Journal Article
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      © 2017 copyright is held by the authors.
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      • Computing and Mathematical Sciences Papers [1454]
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