Minimum maximal matchings in permutahedra

TL;DR

This paper investigates the size of minimal maximal matchings in permutahedra, establishing asymptotic bounds around one-third of the factorial of n, and extends results to products of permutahedra.

Contribution

It provides the first asymptotic bounds for minimal maximal matchings in permutahedra and introduces explicit constructions and bounds for these matchings.

Findings

Minimal size of maximal matchings in permutahedra is asymptotically n!/3.

Established lower bound for M(π_n) as n!(n-1)/(3n-2).

Derived upper bounds using Hall's theorem and explicit constructions.

Abstract

We prove that the minimal size $M(\pi_n)$ of a maximal matching in the permutahedron $\pi_n$ is asymptotically $n!/3$. On the one hand, we obtain a lower bound $M(\pi_n) \ge n! (n-1) / (3n-2)$ by considering $4$-cycles in the permutahedron. On the other hand, we obtain an asymptotical upper bound $M(\pi_n) \le n!(1/3+o(1))$ by multiple applications of Hall's theorem (similar to the approach of Forcade (1973) for the hypercube) and an exact upper bound $M(\pi_n) \le n!/3$ by an explicit construction. We also derive bounds on minimum maximal matchings in products of permutahedra.