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On avoiding odd partial Latin squares and r-multi Latin squares
Journal article   Peer reviewed

On avoiding odd partial Latin squares and r-multi Latin squares

Jaromy Scott Kuhl and Tristan Denley
Discrete Mathematics, Vol.306, pp.2968-2975
306
2006
DOI: https://doi.org/10.1016/j.disc.2006.05.028
Web of Science ID: WOS:000241535800015

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Abstract

We show that for any positive integer k ⩾ 4, if R is a (2k − 1) × (2k − 1) partial Latin square, then R is avoidable given that R contains an empty row, thus extending a theorem of Chetwynd and Rhodes. We also present the idea of avoidability in the setting of partial r-multi Latin squares, and give some partial fillings which are avoidable. In particular, we show that if R contains at most nr/2 symbols and if there is an n × n Latin square L such that δn of the symbols in L cover the filled cells in R where 0<δ<1, then R is avoidable provided r is large enough.

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