Transversals in Latin Arrays with Many Distinct Symbols

Transversals in Latin Arrays with Many Distinct Symbols

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×n array is a selection of n different symbols from different rows and different columns. We prove that every n×n Latin array containing at least (2−2)n2...

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Veröffentlicht in:Journal of combinatorial designs 2018-02, Vol.26 (2), p.84-96
Hauptverfasser: Best, Darcy, Hendrey, Kevin, Wanless, Ian M., Wilson, Tim E., Wood, David R.
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Latin array
Latin square
row‐Latin
Symbols
transversal
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container_issue 2
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container_title Journal of combinatorial designs
container_volume 26
creator Best, Darcy
Hendrey, Kevin
Wanless, Ian M.
Wilson, Tim E.
Wood, David R.
description 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×n array is a selection of n different symbols from different rows and different columns. We prove that every n×n Latin array containing at least (2−2)n2 distinct symbols has a transversal. Also, every n×n row‐Latin array containing at least 14(5−5)n2 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.
doi_str_mv 10.1002/jcd.21566
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subjects Latin array
Latin square
row‐Latin
Symbols
transversal
title Transversals in Latin Arrays with Many Distinct Symbols
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