Geodesics in a Graph of Perfect Matchings

TL;DR

This paper investigates the structure of geodesics in the graph of perfect matchings, providing formulas for distances and counts of shortest paths, and explores properties of non-crossing matchings within this framework.

Contribution

It introduces explicit formulas for the number of geodesics between matchings and analyzes the diameter and antipodal pairs in the graph of perfect matchings.

Findings

Diameter and eccentricity of the graph are m-1.

Number of geodesics between antipodes is m^{m-2}.

Unique maximal number of geodesics between antipodes in non-crossing matchings.

Abstract

Let $\mathscr{P}_{m}$ be the graph on the set of perfect matchings in the complete graph $K_{2m}$, where two perfect matchings are connected by an edge if their symmetric difference is a cycle of length four. This paper studies geodesics in $\mathscr{P}_{m}$. The diameter of $\mathscr{P}_{m}$, as well as the eccentricity of each vertex, are shown to be $m-1$. Two proof are given to show that the number of geodesics between any two antipodes is $m^{m-2}$. The first is a direct proof via a recursive formula, and the second is via reduction to the number of minimal factorizations of a given $m$-cycle in the symmetric group $S_m$. An explicit formula for the number of geodesics between any two matchings in $\mathscr{P}_{m}$ is also given. Let $\mathscr{M}_m$ be the graph on the set of non-crossing perfect matchings of $2m$ labeled points on a circle with the same adjacency condition as in $\mathscr{P}_m$. $\mathscr{M}_m$ is an induced subgraph of $\mathscr{P}_m$, and it is shown that $\mathscr{M}_m$ has exactly one pair of antipodes having the maximal number ($m^{m-2}$) of geodesics between them.