Triangle decompositions of $\lambda K_v-\lambda K_w-\lambda K_u$

Abstract

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 $v\ge w+u+\max \{u,w\}$, $\lambda \left(\left(\genfrac{}{}{0ex}{}{v}{2}\right)-\left(\genfrac{}{}{0ex}{}{u}{2}\right)-\left(\genfrac{}{}{0ex}{}{w}{2}\right)\right)\equiv 0(\mathrm{mod}3)$ and $\lambda (v-w)\equiv \lambda (v-u)\equiv \lambda (v-1)\equiv 0(\mathrm{mod}2)$. 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.