We show that the necessary and sufficient condition for the existence of a balanced bowtie decomposition of the symmetric complete multi-digraph is n 5 and λ(n-1) 0 (mod 6). Decomposition algorithms are also given.

We show that the necessary and sufficient condition for the existence of a balanced C4-trefoil decomposition of the complete multi-graph λKn is λ(n-1) ≡ 0 (mod 24) and n ≤ 10. Decomposition algorithms are also given.

Let t and n be positive integers. We show that the necessary and sufficient condition for the existence of a balanced t-foil decomposition of the complete graph Kn is n 1 (mod 6t). Decomposition algorithms are also given.

We show that the necessary and sufficient condition for the existence of a balanced C4-bowtie decomposition of the complete multi-graph λKn is λ(n - 1) 0 (mod 16) and n 7. 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.

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.

First, we show that the necessary and sufficient condition for the existence of a balanced bowtie decomposition of the complete tripartite multi-graph λ Kn1,n2,n3 is (i) n1=n2=n3 0 (mod 6) for λ 1,5 (mod 6), (ii) n1=n2=n3 0 (mod 3) for λ 2,4 (mod 6), (iii) n1=n2=n3 0 (mod 2) for λ 3 (mod 6), and (iv) n1=n2=n3 2 for λ 0 (mod 6). Next, we show that the necessary and sufficient condition for the existence of a balanced trefoil decomposition of the complete tripartite multi-graph λ Kn1,n2,n3 is (i) n1=n2=n3 0 (mod 9) for λ 1,2,4,5,7,8 (mod 9), (ii) n1=n2=n3 0 (mod 3) for λ 3,6 (mod 9), and (iii) n1=n2=n3 3 for λ 0 (mod 9).