The spectrum of maximal partial Steiner triple systems
type: Item
instance of: scholarly article
title: The spectrum of maximal partial Steiner triple systems
zbMATH Open document ID: 0783.05016
DOI: 10.1007/BF01388482
author: Charles J. Colbourn
author: Alexander Rosa
author: Štefan Znám
published in: Designs, Codes and Cryptography
publication date: 1993-08-29T00:00:00Z
review text: The spectrum \(S^ 3(v)\) is the set of all possible sizes of maximal (that is, nonextendable by adding a triple) partial Steiner triple systems of order \(v\). The maximum value of the spectrum was determined by \textit{Kirkman} [Cambridge and Dublin Math. J. 2, 191-204 (1847)]; the minimum is due to \textit{J. Novák} [Mat. Časopis, Slovensk. Akad. Vied 24, 43-57 (1974; Zbl 0273.05132)]. More recently, \textit{Severn} [C. S. Technical Report \# 172/84, University of Toronto (1984)] completely determined \(S^ 3(v)\) for all odd \(v\); for even \(v\) he determined a major part of the spectrum. In the paper, the authors settle all cases left in doubt by Severn, and thus completely determine the spectrum \(S^ 3(v)\). It is interesting that in all but one case left in doubt before, the values are settled in the negative. The authors' main tool is the application of bounds on the maximum size of a non-bipartite triangle-free graph with all vertices of odd degree.
Mathematics Subject Classification ID: 05B07
zbMATH DE Number: 399363
zbMATH Keywords: maximal partial Steiner triple systems
zbMATH Keywords: spectrum
reviewed by: Jozef Širáň
MaRDI profile type: Publication
cites work: Quadratic leaves of maximal partial triple systems
cites work: Q4017179
cites work: Q5181736
description: scientific article; zbMATH DE number 399363