Pseudo orthogonal Latin squares
type: Item
instance of: scholarly article
title: Pseudo orthogonal Latin squares
zbMATH Open document ID: 1461.05017
author: Shahab Faruqi
author: S. A. Katre
author: Manisha Garg
published in: Discrete Mathematics and Applications
publication date: 2021-02-26T00:00:00Z
review text: The authors prove that the existence of a family of \(m\) mutually pseudo orthogonal Latin squares of order \(n\) (\(\mathrm{POLS}(m;n)\)) is equivalent to the existence of a family of \(m\) mutually orthogonal Latin squares of order \(n\) (\(\mathrm{MOLS}(m;n)\)). Two distinct Latin squares \(A\), \(B\) of order \(n\) are said to be pseudo orthogonal if for each \(1\leq i\), \(j\leq n\) there is a \(k\) with \(1\leq k\leq n\) satisfying \(A(i,k)=B(j,k)\). The notion of pseudo orthogonality was first introduced in [\textit{N. S. Bolshakova}, Discrete Math. Appl. 18, No. 2, 187--197 (2008; Zbl 1200.05141); translation from Diskretn. Mat. 20, No. 1, 70--79 (2008)]. The authors use graph invariants to establish a relationship between pseudo orthogonal Latin squares and orthogonal Latin squares. Specifically, for a graph \(G\), these invariants are the clique partition number of \(G\) and the size \(\pi (G)\) of a prime set for an optimum \(\pi\)-labeling of \(G\). For \(m\geq 0\) and \(n\geq 2\) let \(G\) be the graph obtained by taking the union of \(m+2\) vertex disjoint complete graphs on \(n\) vertices. The authors' theorem asserting the equivalence of a family of \(\mathrm{MOLS}(m;n)\) and a family of \(\mathrm{POLS}(m;n)\) also has two intermediate equivalent statements: (1) \(\pi (G)=n^2\) and (2) there is a clique partition of \(\bar G\) which consists only of \(K_{m+2}\)'s. The authors apply this theorem and surrounding ideas to obtain exact values of clique partition numbers of several classes of complete multipartite graphs and tensor products of complete graphs.
Mathematics Subject Classification ID: 05B15
Mathematics Subject Classification ID: 05C69
Mathematics Subject Classification ID: 05C78
zbMATH DE Number: 7316009
zbMATH Keywords: mutually orthogonal Latin squares
zbMATH Keywords: mutually pseudo orthogonal Latin squares
zbMATH Keywords: clique partition numbers
zbMATH Keywords: \(\pi\) labeling
Wikidata QID: Q113264444
reviewed by: John Lorch
MaRDI profile type: Publication
full work available at URL: dma-2021-0002
OpenAlex ID: W3130426263
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DOI: 10.1515/DMA-2021-0002
description: scientific article; zbMATH DE number 7316009