Partitions of sets of two-fold triple systems, and their relations to some strongly regular graphs
type: Item
instance of: scholarly article
title: Partitions of sets of two-fold triple systems, and their relations to some strongly regular graphs
zbMATH Open document ID: 0837.05021
DOI: 10.1007/BF01787815
author: Anne Penfold Street
author: Rudolf Mathon
published in: Graphs and Combinatorics
publication date: 1996-02-13T00:00:00Z
review text: A two-fold triple system (TTS) is a 2-\((v,k,2)\) design (and hence \(v \equiv 1\) or \(3 \pmod 6\)). An overlarge set of TTS, denoted by \(\text{OS(TTS} (v))\) is a set of \(v + 1\) mutually disjoint \(\text{TTS} (v)\) (i.e. they have no triple in common), each of them based on a different \(v\)-subset of a set \(X\) of cardinality \(v + 1\). The authors prove that if \(|X |= 7\), then the 84 distinct (but all isomorphic) TTS(6) based on \(X\), can be partitioned into 12 OS(TTS(6)) each containing 7 TTS(6). Moreover there exists a resolution into 84 parallel classes of the set of 1008 distinct OS(TTS(6)) based on \(X\). Using this terminology they revisit the Hoffman-Singleton graph as well as the Higman-Sims graph. They proof, using their terminology, the well-known fact that this last graph is the union of two Hoffman-Singleton graphs. The reviewer disagrees with the authors that their description of the Higman-Sims graph is the first elementary construction.
Mathematics Subject Classification ID: 05B07
Mathematics Subject Classification ID: 05E30
zbMATH DE Number: 837841
zbMATH Keywords: strongly regular graphs
zbMATH Keywords: triple system
zbMATH Keywords: Hoffman-Singleton graph
zbMATH Keywords: Higman- Sims graph
describes a project that uses: nauty
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf01787815
OpenAlex ID: W1990840290
description: scientific article; zbMATH DE number 837841