We show that the necessary and sufficient condition for the existence of a balanced quatrefoil decomposition of the complete multigraph λKn is n 9 and λ(n - 1) 0 (mod 24). Decomposition algorithms are also given.

We show that the necessary and sufficient condition for the existence of a balanced bowtie decomposition of the complete multigraph λKn is n 5 and λ(n-1) 0 (mod 12). Decomposition algorithms are also given.