Deranged Perfect Matchings on complete graph and balanced complete r-partite graph
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2025
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| _version_ | 1866910062147010560 |
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| author | Deng, Boqing |
| author_facet | Deng, Boqing |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_13799 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Deranged Perfect Matchings on complete graph and balanced complete r-partite graph Deng, Boqing Combinatorics 05C30, 05C70 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. |
| title | Deranged Perfect Matchings on complete graph and balanced complete r-partite graph |
| topic | Combinatorics 05C30, 05C70 |
| url | https://arxiv.org/abs/2505.13799 |