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We say that two graph $G$ and $H$ with the same vertex set commute if their adjacency matrices commute. In this article, we show that for any natural number $r$, the complete multigraph $K^{r}_{n}$ is decomposable into commuting perfect matchings if and only if $n$ is a 2-power. Also, it is shown that the complete graph $K_{n}$ is decomposable into commuting Hamilton cycles if and only if $n$ is a prime number.
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