Erd\H{o}s-Ko-Rado for Perfect Matchings

TL;DR

This paper provides an algebraic proof of the maximum size of intersecting families of perfect matchings in complete graphs, characterizes extremal families, and explores related algebraic structures from a symmetric association scheme.

Contribution

It offers a new algebraic proof of a known combinatorial bound and investigates the structure of non-Hamiltonian families using association schemes.

Findings

Maximum intersecting family size is (2(n-1) - 1)!!

Extremal families are those containing a fixed edge

Non-Hamiltonian families also achieve this maximum size

Abstract

A perfect matching of a complete graph $K_{2n}$ is a 1-regular subgraph that contains all the vertices. Two perfect matchings intersect if they share an edge. It is known that if $\mathcal{F}$ is family of intersecting perfect matchings of $K_{2n}$, then $|\mathcal{F}| \leq (2(n-1) - 1)!!$ and if equality holds, then $\mathcal{F} = \mathcal{F}_{ij}$ where $ \mathcal{F}_{ij}$ is the family of all perfect matchings of $K_{2n}$ that contain some fixed edge $ij$. We give a short algebraic proof of this result, resolving a question of Godsil and Meagher. Along the way, we show that if a family $\mathcal{F}$ is non-Hamiltonian, that is, $m \cup m' \not \cong C_{2n}$ for any $m,m' \in \mathcal{F}$, then $|\mathcal{F}| \leq (2(n-1) - 1)!!$ and this bound is met with equality if and only if $\mathcal{F} = \mathcal{F}_{ij}$. Our results make ample use of a somewhat understudied symmetric commutative association scheme arising from the Gelfand pair $(S_{2n},S_2 \wr S_n)$. We give an exposition of a few new interesting objects that live in this scheme as they pertain to our results.