Balanced bowtie and trefoil decomposition of complete tripartite multigraphs

First, we show that the necessary and sufficient condition for the existenc e of a balanced bowtie decomposition of the complete tripartite multi-graph lambdaK(n1,n2,n3) is (i) n(1) = n(2) = n(3) = 0 (mod 6) for lambda = 1,5 ( mod 6), (ii) n(1) = n(2) = n(3) = 0 (mod 3) for lambda = 2,4 (mod 6), (iii) n(1) = n(2) = n(3) = 0 (mod 2) for lambda = 3 (mod 6), and (iv) n(1) = n(2 ) = n(3) greater than or equal to 2 for lambda = 0 (mod 6). Next, we show t hat the necessary and sufficient condition for the existence of a balanced trefoil decomposition of the complete tripartite multi-graph lambdaK(n1,n2, n3) is (i) n(1) = n(2) = n(3) = 0 (mod 9) for lambda = 1, 2, 4, 5, 7, 8 (mo d 9), (ii) n(1) = n(2) = n(3) = 0 (mod 3) for lambda = 3, 6 (mod 9), and (i ii) n(1) = n(2) = n(3) greater than or equal to 3 for lambda = 0 (mod 9).