Decomposition of \(\lambda K_{v}\) into Multigraphs with Four Vertices and Five Edges

We give necessary and sufficient conditions to decompose \( \lambda \) copies, where necessarily \( \lambda \geq 2 \), of the complete graph \( K_v \), into so-called “2-petal”, “stem-infinity”, “barbell”, and “box-edge” graphs, all with four vertices and five edges.