Deranged Perfect Matchings on complete graph and balanced complete r-partite graph

We proved that for any finite collection of sparse subgraphs $(D_m{)}_{m=1}^{\mathrm{\ell }}$ of the complete graph $K_{2n}$, and a uniformly chosen perfect matching $R$ in $K_{2n}$, the random vector $(\mathrm{\mid }E(R\cap D_m)\mathrm{\mid }{)}_{m=1}^{\mathrm{\ell }}$ jointly converges to a vector of independent Poisson random variables with mean $\mathrm{\mid }E(D_m)\mathrm{\mid }\mathrm{/}(2n)$. We also showed a similar result when $K_{2n}$ is replaced by the balanced complete $r$-partite graph $K_{r\times 2n\mathrm{/}r}$ for fixed $r$ and determined the asymptotic joint distribution. The proofs rely on elementary tools of the Principle of Inclusion-Exclusion and generating functions. These results extend recent works of Johnston, Kayll and Palmer, Spiro and Surya, and Granet and Joos from the univariate to the multivariate setting.