Existence of perfect Mendelsohn designs with \(k=5\) and \(\lambda{}>1\)
type: Item
instance of: scholarly article
title: Existence of perfect Mendelsohn designs with \(k=5\) and \(\lambda{}>1\)
zbMATH Open document ID: 0756.05011
DOI: 10.1016/0012-365X(92)90263-F
author: S. Singh
published in: Discrete Mathematics
publication date: 1993-01-16T00:00:00Z
review text: Let \(v,k\), and \(\lambda\) be positive integers. A \((v,k,\lambda)\)- Mendelsohn design (briefly \((v,k,\lambda)-\text{MD})\) is a pair \((X,{\mathcal B})\) where \(X\) is a \(v\)-set (of points) and \({\mathcal B}\) is a collection of cyclically ordered \(k\)-subsets of \(X\) (called blocks) such that every ordered pair of points of \(X\) is consecutive in exactly \(\lambda\) blocks of \({\mathcal B}\). A set of \(k\) distinct elements \(\{a_ 1,a_ 2,\dots,a_ k\}\) is said to be cyclically ordered by \(a_ 1<a_ 2<\cdots<a_ k<a_ 1\) and the pair \(a_ i\), \(a_{i+t}\) is said to be \(t\)-apart in cyclic \(k\)-tuple \((a_ 1,a_ 2,\dots,a_ k)\) where \(i+t\) is taken modulo \(k\). If for all \(t=1,2,\dots,k-1\), every ordered pair of points of \(X\) is \(t\)-apart in exactly \(\lambda\) blocks of \({\mathcal B}\), then the \((v,k,\lambda)\)-MD is called a perfect design and is denoted briefly by \((v,k,\lambda)\)-PMD. This paper is concerned mainly with the case where \(k=5\) and \(\lambda>1\). It is shown that the necessary condition for the existence of a \((v,5,\lambda)\)-PMD, namely, \(\lambda v(v-1)\equiv 0\pmod 5\), is also sufficient for \(\lambda>1\) with the possible exception of pairs \((v,\lambda)\) where \(\lambda=5\) and \(v=18\) or 28.
Mathematics Subject Classification ID: 05B05
Mathematics Subject Classification ID: 05B30
zbMATH DE Number: 89919
zbMATH Keywords: Mendelsohn designs
zbMATH Keywords: perfect design
zbMATH Keywords: blocks
zbMATH Keywords: cyclically ordered
reviewed by: Ratnakaram Nava Mohan
MaRDI profile type: Publication
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full work available at URL: 0012-365x(92)90263-f
OpenAlex ID: W2000317716
description: scientific article; zbMATH DE number 89919