Bicyclic Steiner triple systems
type: Item
instance of: scholarly article
title: Bicyclic Steiner triple systems
zbMATH Open document ID: 0797.05015
DOI: 10.1016/0012-365X(94)90102-3
author: Rebecca Calahan-Zijlstra
author: Robert B. Gardner
published in: Discrete Mathematics
publication date: 1994-10-10T00:00:00Z
review text: A permutation \(\pi\) of a \(v\)-element set is said to be of type \([\pi]=[p_ 1,p_ 2,\ldots,p_ v]\) if the disjoint cyclic decomposition of \(\pi\) contains \(p_ i\) cycles of length \(i\). Thus a cyclic Steiner triple system STS\((v)\) is one admitting an automorphism of type \([0,0,\ldots,0,1]\). A bicyclic Steiner triple system is defined to be one admitting an automorphism of type \([0,0,\ldots,0,p_{N_ 1},0,\ldots,0,p_{N_ 2},0,\ldots,0]\) where \(p_{N_ 1}=p_{N_ 2}=1\), \(N_ 1<N_ 2\) and \(N_ 1+N_ 2=v\). In this paper the authors give necessary and sufficient conditions for the existence of bicyclic Steiner triple systems. The case where \(N_ 1=1\) (a 1-rotational STS) was dealt with by \textit{K. T. Phelps} and \textit{A. Rosa} [Discrete Math. 33, 57-66 (1981; Zbl 0463.05014)]. When \(N_ 1>1\), the authors show that necessary and sufficient conditions for existence of a bicyclic STS are that \(N_ 1\equiv 1\) or \(3\pmod 6\), \(N_ 1\neq 9\), \(N_ 1| N_ 2\), and of course \(v=N_ 1+N_ 2\equiv 1\) or \(3\pmod 6\).
Mathematics Subject Classification ID: 05B07
zbMATH DE Number: 562581
zbMATH Keywords: reverse STS
zbMATH Keywords: \(k\)-rotational STS
zbMATH Keywords: Steiner triple system
zbMATH Keywords: bicyclic Steiner triple system
reviewed by: Elizabeth J. Billington
MaRDI profile type: Publication
cites work: Rotational Steiner triple systems
cites work: A note on reverse Steiner triple systems
cites work: Q4288161
cites work: Verification of a Conjecture of Th. Skolem.
cites work: Steiner triple systems with rotational automorphisms
cites work: On reverse Steiner triple systems
cites work: On certain distributions of integers in pairs with given differences
cites work: The existence of reverse Steiner triple systems
cites work: A simplification of the proof of the existence of reverse Steiner triple systems of order congruent to 1 modulo 24
full work available at URL: 0012-365x(94)90102-3
OpenAlex ID: W2028767914
description: scientific article; zbMATH DE number 562581