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arXiv · 2505.13799

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

Boqing Deng
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Abstract

We proved that for any finite collection of sparse subgraphs $(D_m)_{m=1}^\ell$ of the complete graph $K_{2n}$, and a uniformly chosen perfect matching $R$ in $K_{2n}$, the random vector $(|E(R \cap D_m)|)_{m=1}^\ell$ jointly converges to a vector of independent Poisson random variables with mean $|E(D_m)|/(2n)$. We also showed a similar result when $K_{2n}$ is replaced by the balanced complete $r$-partite graph $K_{r \times 2n/r}$ 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.

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Boqing Deng. 2025-05-20. Deranged Perfect Matchings on complete graph and balanced complete r-partite graph. https://doi.org/10.7151/dmgt.2610

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