Bicyclic Steiner triple systems
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 \(3\pmod 6\).
scientific article; zbMATH DE number 562581