✦ LIBER ✦

Latin squares with restricted transversals

✍ Scribed by Judith Egan; Ian M. Wanless

Publisher: John Wiley and Sons
Year: 2011
Tongue: English
Weight: 149 KB
Volume: 20
Category: Article
ISSN: 1063-8539
DOI: 10.1002/jcd.20297

No coin nor oath required. For personal study only.

✦ Synopsis

Abstract

We prove that for all odd m≥3 there exists a latin square of order 3 m that contains an (m−1) × m latin subrectangle consisting of entries not in any transversal. We prove that for all even n≥10 there exists a latin square of order n in which there is at least one transversal, but all transversals coincide on a single entry. A corollary is a new proof of the existence of a latin square without an orthogonal mate, for all odd orders n≥11. Finally, we report on an extensive computational study of transversal‐free entries and sets of disjoint transversals in the latin squares of order n⩽9. In particular, we count the number of species of each order that possess an orthogonal mate. © 2011 Wiley Periodicals, Inc. J Combin Designs 20:124‐141, 2012

📜 SIMILAR VOLUMES

Latin squares with one subsquare

✍ I. M. Wanless 📂 Article 📅 2001 🏛 John Wiley and Sons 🌐 English ⚖ 207 KB

Latin squares with no small odd plexes

✍ Judith Egan; Ian M. Wanless 📂 Article 📅 2008 🏛 John Wiley and Sons 🌐 English ⚖ 242 KB

## Abstract A __k__‐plex in a Latin square of order __n__ is a selection of __kn__ entries in which each row, column, and symbol is represented precisely __k__ times. A transversal of a Latin square corresponds to the case __k__ = 1. We show that for all even __n__ > 2 there exists a Latin square o

Bachelor latin squares with large indivi

Bachelor latin squares with large indivisible plexes

✍ Judith Egan 📂 Article 📅 2011 🏛 John Wiley and Sons 🌐 English ⚖ 116 KB

In a latin square of order n, a k-plex is a selection of kn entries in which each row, column, and symbol occurs k times. A 1-plex is also called a transversal. A k-plex is indivisible if it contains no c-plex for 0<c<k. We prove that, for all n ≥ 4, there exists a latin square of order n that can b

TRANSVERSE VIBRATIONS OF A SQUARE MEMBRA

TRANSVERSE VIBRATIONS OF A SQUARE MEMBRANE WITH AN ECCENTRIC CIRCULAR OR QUASI-SQUARE HOLE

✍ R.H. Gutierrez; P.A.A. Laura 📂 Article 📅 1997 🏛 Elsevier Science 🌐 English ⚖ 194 KB

TRANSVERSE VIBRATIONS OF A CIRCULAR PLAT

TRANSVERSE VIBRATIONS OF A CIRCULAR PLATE WITH A CONCENTRIC SQUARE HOLE WITH FREE EDGES

✍ D.R. Avalos; H.A. Larrondo; P.A.A. Laura; V. Sonzogni 📂 Article 📅 1998 🏛 Elsevier Science 🌐 English ⚖ 112 KB

Negative Latin square type partial diffe

Negative Latin square type partial difference sets and amorphic association schemes with Galois rings

✍ John Polhill 📂 Article 📅 2009 🏛 John Wiley and Sons 🌐 English ⚖ 176 KB

## Abstract A partial difference set (PDS) having parameters (__n__^2^, __r__(__n__−1), __n__+__r__^2^−3__r, r__^2^−__r__) is called a __Latin square type__ PDS, while a PDS having parameters (__n__^2^, __r__(__n__+1), −__n__+__r__^2^+3__r, r__^2^ +__r__) is called a __negative Latin square type__