Cyclic Steiner Quadruple Systems and Köhler's orbit graphs

Abstract

In this article we are concerned with the problem of the existence of strictly cyclic Steiner Quadruple Systems sSQS(v), where v ≡ 2, 10 (24). E. Köhler (cf. (Köhler 1978)) used an orbit graph approach to handle such systems and obtained the result that in case p is a prime number with p ≡ 53, 77 (120) then sSQS(v) exists provided that the associated orbit graph OKG(p) is bridgeless. We continue these investigations by classifying the orbit graphs OKG(p) with p ≡ 5 (12), where the ones with p ≡ 53, 77 (120) constitute one out of four classes and thus show that sSQS(2p), p ≡ 5 (12) exists if OKG(p) or a reduced graph of it is bridgeless by discussing the four classes separately. Subsequent to this discussion we use the proof of Theorem 2 (Siemon 1991) to state that the bridgelessness of the graphs in all classes is equivalent to the number theoretic claim (3.1).