Latin squares with one subsquare
type: Item
instance of: scholarly article
zbMATH Open document ID: 0971.05029
author: Ian M. Wanless
publication date: 2001-10-21T00:00:00Z
Mathematics Subject Classification ID: 05B15
zbMATH DE Number: 1595942
zbMATH Keywords: Latin square
zbMATH Keywords: subsquare
zbMATH Keywords: skip square
zbMATH Keywords: corrupting pair
zbMATH Keywords: corrupted product
MaRDI profile type: Publication
cites work: The Existence of a Subsquare Free Latin Square of Side 12
cites work: Latin squares with no proper subsquares
cites work: On certain constructions for latin squares with no latin subsquares of order two
cites work: Most Latin squares have many subsquares
cites work: THE 7 × 7 SQUARES
cites work: An Omission in Norton's List of $7 \times 7$ Squares
OpenAlex ID: W1998397814
title: Latin squares with one subsquare
DOI: 10.1002/1520-6610(2001)9:2%3C128::AID-JCD1003%3E3.0.CO;2-C
full work available at URL: https://doi.org/10.1002/1520-6610(2001)9:2%3C128::aid-jcd1003%3E3.0.co
published in: Journal of Combinatorial Designs
review text: A subsquare of a Latin square is a submatrix which is itself a Latin square. A subsquare of a Latin square of order \(n\) is proper if its order \(m\) satisfies \(1<m<n\). A Latin square without any proper subsquares is said to be \(N_\infty\). This paper is a continuation of the quest to completely determine the spectrum of \(N_\infty\) Latin squares. The best general result in this area to date is that of \textit{L. D. Andersen} and \textit{E. Mendelsohn} in [A direct construction for Latin squares without proper subsquares, Ann. Discrete Math. 15, 27-53 (1982; Zbl 0493.05006)] who give a construction for \(N_\infty\) Latin squares of orders not of the form \(2^a3^b\). The rationale in this paper is to examine methods for constructing Latin squares which have exactly one subsquare (this class is called \(\mathcal{U}\)), and to then apply a small perturbation to destroy the existing subsquare. This idea was used by the author in his PhD thesis (Australian National University, 1997) to create \(N_\infty\) squares of orders \(32\), \(64\) and \(128\). In fact the author extended this idea to construct \(N_\infty\) squares for all previously unknown orders of the form \(2^a3^b < 256\). Two classes of Latin squares in \(\mathcal{U}\) are examined here. The first class consists of squares obtained by a prolongation of a skip square using transversals of a particular form. This class includes known squares due to \textit{M. McLeish} in [A direct construction of Latin squares with no subsquares of order two, Ars Comb. 10, 179-186 (1980; Zbl 0466.05013)] and to \textit{A. Kotzig} and \textit{J. Turgeon} [On certain constructions for Latin squares with no Latin subsquares of order two, Discrete Math. 16, 263-270 (1976; Zbl 0351.05016)]. The author prescribes conditions under which the resulting square belongs to \(\mathcal{U}\). In particular he shows that the order \(m\) of the proper subsquare of a Latin square of order \(n\) in \(\mathcal{U}\) must satisfy \(2m+1 \leq n\…
reviewed by: Peter B. Gibbons
description: scientific article