Avoiding pairs of partial Latin squares
type: Item
instance of: scholarly article
title: Avoiding pairs of partial Latin squares
zbMATH Open document ID: 1308.05029
author: Hannah Hinojosa
author: Jaromy Kuhl
published in: Graphs and Combinatorics
publication date: 2014-06-16T00:00:00Z
review text: A set of partial Latin squares of order \(n\) is avoidable if there is a Latin square of order \(n\) that does not agree with any of the partial Latin squares in any cell. The authors show that if \(k>m^3(m^2-1)/2\) and \(m\geq4\) then any two partial Latin squares of order \(mk\) are avoidable (they also improve this result somewhat when \(m=4\)). When the paper was written, this was the only result showing avoidability of more than one partial Latin square of odd order. However, during the years that the paper took to appear it has been superseded by the much more general results of \textit{L. J. Andrén} et al. [Comb. Probab. Comput. 22, No. 2, 184--212 (2013; Zbl 1260.05020)]. A natural context where avoidability of Latin squares arises is in adding layers to Latin cuboids, trying to build a Latin cube. In this context, it is known [\textit{D. Bryant} et al., SIAM J. Discrete Math. 26, No. 1, 239--249 (2012; Zbl 1246.05027)] that there are sets of \(n/2+O(1)\) Latin squares of order \(n\) that are not avoidable. For partial Latin squares, it is possible [\textit{J. Cutler} and \textit{L.-D. Öhman}, Electron. J. Comb. 13, No. 1, Research paper R47, 9 p. (2006; Zbl 1098.05016)] to find sets of size \(n/3+O(1)\) that are unavoidable. There is still a large gap between these constructions and the results mentioned above.
Mathematics Subject Classification ID: 05B15
zbMATH DE Number: 6304677
zbMATH Keywords: partial Latin square
zbMATH Keywords: avoidable
reviewed by: Ian M. Wanless
MaRDI profile type: Publication
full work available at URL: s00373-013-1301-4
OpenAlex ID: W2065132880
cites work: Q3103895
cites work: Nonextendible Latin cuboids
cites work: On avoiding some families of arrays
cites work: Q3114964
cites work: Avoiding multiple entry arrays
cites work: Avoiding partial Latin squares and intricacy
cites work: Latin squares with forbidden entries
cites work: Degrees giving independent edges in a hypergraph
cites work: A few remarks on avoiding partial Latin squares.
cites work: Q4677944
cites work: Partial Latin squares are avoidable
DOI: 10.1007/S00373-013-1301-4
description: scientific article; zbMATH DE number 6304677