Embedding partial extended triple systems and totally symmetric quasigroups
type: Item
instance of: scholarly article
title: Embedding partial extended triple systems and totally symmetric quasigroups
zbMATH Open document ID: 0901.05011
DOI: 10.1016/S0012-365X(96)00297-X
author: C. A. Rodger
author: Michael Raines
published in: Discrete Mathematics
publication date: 1998-11-26T00:00:00Z
review text: Let \(K^+_n\) be the complete graph on \(n\) vertices with exactly one loop incident with each vertex (a loop is considered an edge). An extended triple is defined to be either a loop, a loop with an edge attached, or a copy of \(K_3\). A (partial) extended triple system of order \(n\) is an ordered pair \((V, B)\), where \(B\) is a set of extended triples defined on the vertex set \(V\) which partitions (a subset of) the edges of \(K^+_n\). Partial extended triple systems and extended triple systems of order \(n\) are denoted \(\text{PETS}(n)\) and \(\text{ETS}(n)\), respectively. A \(\text{PETS}(V, B)\) is said to be embedded in an \(\text{ETS}(V', B')\) if \(V\subseteq V'\) and \(B\subseteq B'\). In 1987, Hoffman and Rodger proved that an \(\text{ETS}(n)\) can be embedded in an \(\text{ETS}(v)\) if and only if \(v\geq 2n\), \(v\) is even if \(n\) is even, and \((v, n)\neq (6k+ 5, 12k+ 12)\). The main result of this paper is that a \(\text{PETS}(n)\) can be embedded in can \(\text{EPS}(v)\) for any \(v\geq 4u+ 6\) with \(v\equiv 2\text{ mod }4\).
reviewed by: J. H. Dinitz
Mathematics Subject Classification ID: 05B07
zbMATH DE Number: 1109969
zbMATH Keywords: complete graph
zbMATH Keywords: extended triple system
MaRDI profile type: Publication
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full work available at URL: s0012-365x(96)00297-x
OpenAlex ID: W2087406041
description: scientific article; zbMATH DE number 1109969