Triangle decompositions of λKv−λKw−λKu\lambda K_v-\lambda K_w-\lambda K_u

Denote by $\lambda K_v$ the complete graph of order $v$ with multiplicity $\lambda$. Let $\lambda K_v-\lambda K_w-\lambda K_u$ be the graph obtained from $\lambda K_v$ by the removal of the edges of two vertex disjoint complete multi-subgraphs with multiplicity $\lambda$ of orders $w$ and $u$, respectively. When $\lambda$ is odd, it is shown that there exists a triangle decomposition of $\lambda K_v-\lambda K_w-\lambda K_u$ if and only if $v\geq w+u+\max\{u,w\}$, $\lambda \left({v\choose 2}-{u\choose 2}-{w\choose 2}\right) \equiv 0 \pmod 3$ and $\lambda (v-w) \equiv \lambda (v-u) \equiv \lambda (v-1) \equiv 0 \pmod 2$. When $\lambda$ is even, it is shown that for large enough $v$, the elementary necessary conditions for the existence of a triangle decomposition of $\lambda K_v-\lambda K_w-\lambda K_u$ are also sufficient