An algebraic proof of the Erd\H{o}s-Ko-Rado theorem for intersecting families of perfect matchings

TL;DR

This paper provides an algebraic proof of the Erd ext{"o}s-Ko-Rado theorem specifically for perfect matchings, identifying the maximum intersecting families as those containing a fixed edge.

Contribution

It introduces an algebraic approach to prove the maximum intersecting family of perfect matchings, utilizing eigenvalues and properties of the perfect matching polytope.

Findings

Identifies the largest intersecting family as matchings containing a fixed edge

Determines the least eigenvalue of the perfect matching derangement graph

Shows the perfect matching derangement graph is not a Cayley graph

Abstract

In this paper we give a proof that the largest set of perfect matchings, in which any two contain a common edge, is the set of all perfect matchings that contain a fixed edge. This is a version of the famous Erd\H{o}s-Ko-Rado theorem for perfect matchings. The proof given in this paper is algebraic, we first determine the least eigenvalue of the perfect matching derangement graph and use properties of the perfect matching polytope. We also prove that the perfect matching derangement graph is not a Cayley graph.