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

TL;DR

The paper proves that for a fixed finite collection of sparse subgraphs $(D_m)_{m=1}^{\ell}$, the vector $(|E(R\cap D_m)|)_{m=1}^{\ell}$, with $R$ a uniformly random perfect matching, converges to a multivariate Poisson distribution. For $G=K_{2n}$ the limiting means are $|E(D_m)|/(2n)$ and, when the $D_m$ are disjoint, the coordinates become asymptotically independent; for the balanced complete $r$-partite graph $K_{r\times (r-1)n}$ the means are $|E(D_m)|/((r-1)^2 n)$ with the same independence in the disjoint case. A decomposition argument handles non-disjoint $D_m$ by splitting into disjoint Poisson components and summing. The results are proved via the Principle of Inclusion-Exclusion and generating functions, complemented by a Tannery-type limit argument, thereby extending prior univariate Poisson-approximation results to the multivariate setting and providing a macroscopic justification for Poisson heuristics in deranged matchings on large graphs.

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.

Theorems & Definitions (38)

• Definition 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 2.1: Principle of Inclusion-Exclusion
• proof
• Definition 2.2
• Corollary 2.3
• Theorem 2.4: Tannery's Theorem
• Lemma 2.5
• proof