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arXiv · 1612.09443

Transversals in Latin arrays with many distinct symbols

Darcy Best·Kevin Hendrey·Ian M. Wanless·Tim E. Wilson·David R. Wood
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Abstract

An array is row-Latin if no symbol is repeated within any row. An array is Latin if it and its transpose are both row-Latin. A transversal in an $n\times n$ array is a selection of $n$ different symbols from different rows and different columns. We prove that every $n \times n$ Latin array containing at least $(2-\sqrt{2}) n^2$ distinct symbols has a transversal. Also, every $n \times n$ row-Latin array containing at least $\frac14(5-\sqrt{5})n^2$ distinct symbols has a transversal. Finally, we show by computation that every Latin array of order $7$ has a transversal, and we describe all smaller Latin arrays that have no transversal.

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Darcy Best, Kevin Hendrey, Ian M. Wanless, Tim E. Wilson, David R. Wood. 2016-12-30. Transversals in Latin arrays with many distinct symbols. https://doi.org/10.1002/jcd.21566

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