Bicyclic decompositions of Kv into copies of K3 ∪{e}

Abstract

A decomposition of the complete graph on v vertices, K_v, into copies of K₃ with a pendant edge is called a "lollipop" system of order v, denoted LS(v). We give necessary and sufficient conditions for the existence of a LS(v) admitting an automorphism consisting of two disjoint cycles. We also give a brief proof that the previously known sufficient conditions for the existence of a cyclic LS(v) are in fact necessary.