Monogamous latin squares

We show for all n ∉ { 1 , 2 , 4 } that there exists a latin square of order n that contains two entries γ 1 and γ 2 such that there are some transversals through γ 1 but they all include γ 2 as well. We use this result to show that if n > 6 and n is not of the form 2 p for a prime p ⩾ 11 then the...

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Bibliographic Details
Published in:Journal of combinatorial theory. Series A 2011-04, Vol.118 (3), p.796-807
Main Authors: Danziger, Peter, Wanless, Ian M., Webb, Bridget S.
Format: Article
Language:English
Subjects:
Latin square
maxMOLS
MOLS
Monogamous square
Transversal
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Summary:We show for all n ∉ { 1 , 2 , 4 } that there exists a latin square of order n that contains two entries γ 1 and γ 2 such that there are some transversals through γ 1 but they all include γ 2 as well. We use this result to show that if n > 6 and n is not of the form 2 p for a prime p ⩾ 11 then there exists a latin square of order n that possesses an orthogonal mate but is not in any triple of MOLS. Such examples provide pairs of 2-maxMOLS.
ISSN:0097-3165
1096-0899
DOI:10.1016/j.jcta.2010.11.011