Monogamous latin squares

Danziger, Peter; Wanless, Ian M. and Webb, Bridget S. (2011). Monogamous latin squares. Journal of Combinatorial Theory, Series A, 118(3) pp. 796–807.

DOI: https://doi.org/10.1016/j.jcta.2010.11.011

Abstract

We show for all n not in {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 2p for a prime p greater or equal to 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.

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