Constructions of perfect Mendelsohn designs
type: Item
instance of: scholarly article
title: Constructions of perfect Mendelsohn designs
zbMATH Open document ID: 0756.05012
DOI: 10.1016/0012-365X(92)90264-G
author: S. Singh
published in: Discrete Mathematics
publication date: 1993-01-16T00:00:00Z
review text: Let \(n\) and \(k\) be positive integers. An \((n,k,1)\)-Mendelsohn design is an ordered pair \((V,{\mathcal C})\) where \(V\) is the vertex set of \(D_ n\), the complete directed graph on \(n\) vertices, and \({\mathcal C}\) is a set of directed cycles (called blocks) of length \(k\) which form an arc-disjoint decomposition of \(D_ n\). An \((n,k,1)\)-Mendelsohn design is called a perfect design and denoted briefly by \((n,k,1)\)-PMD if for any \(r\), \(1\leq r\leq k-1\), and for each \((x,y)\in V\times V\) there is exactly one cycle \(c\in{\mathcal C}\) in which the (directed) distance along \(c\) from \(x\) to \(y\) is \(r\). A necessary condition for the existence of an \((n,k,1)\)-PMD is \(n(n-1)\equiv 0\pmod k\). In this paper some new techniques used in the construction of PMD's, including constructions of the product type, are described. As an application, it is shown that the necessary condition for the existence of an \((n,5,1)\)-PMD is also sufficient, except for \(n=6\) and with at most 21 possible exceptions of \(n\) of which 286 is the largest.
Mathematics Subject Classification ID: 05B05
Mathematics Subject Classification ID: 05B07
Mathematics Subject Classification ID: 05B30
zbMATH DE Number: 89920
zbMATH Keywords: Mendelsohn designs
zbMATH Keywords: complete directed graph
zbMATH Keywords: blocks
zbMATH Keywords: decomposition
zbMATH Keywords: perfect design
reviewed by: Ratnakaram Nava Mohan
MaRDI profile type: Publication
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Wikidata QID: Q126409327
full work available at URL: 0012-365x(92)90264-g
OpenAlex ID: W1994566808
description: scientific article; zbMATH DE number 89920