Bicyclic decompositions of Kv into copies of K3 ∪e - Gardner, Robert B. (Utilitas Mathematica, 1998)
A decomposition of the complete graph on v vertices, Kv, 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.