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Minimum maximal matchings in permutahedra

Sofia Brenner, Jiří Fink, Hung. P. Hoang, Arturo Merino, Vincent Pilaud

TL;DR

This work determines the asymptotics of the minimal size of a maximal matching in permutahedra: $\mathcal{M}(\Pi_n) \sim n!/3$. It provides a general 4-cycle-based lower bound that yields $\mathcal{M}(\Pi_n) \ge n!(n-1)/(3n-2)$ and complements it with an asymptotically tight upper bound derived from Hall's theorem, together with a constructive proof that achieves $\mathcal{M}(\Pi_n) \le n!/3$. The authors give an explicit, scalable construction for the maximal matching, prove its maximality, and extend the analysis to Cartesian products of permutahedra, obtaining matching upper and lower bounds in that broader class. These results not only resolve the permutahedron case up to lower-order terms but also connect to hypercube techniques and broader bipartite graph frameworks. The paper concludes with open problems, notably a symmetric chain decomposition for $\Pi_n$ and extensions to other polytopes.

Abstract

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

Minimum maximal matchings in permutahedra

TL;DR

This work determines the asymptotics of the minimal size of a maximal matching in permutahedra: . It provides a general 4-cycle-based lower bound that yields and complements it with an asymptotically tight upper bound derived from Hall's theorem, together with a constructive proof that achieves . The authors give an explicit, scalable construction for the maximal matching, prove its maximality, and extend the analysis to Cartesian products of permutahedra, obtaining matching upper and lower bounds in that broader class. These results not only resolve the permutahedron case up to lower-order terms but also connect to hypercube techniques and broader bipartite graph frameworks. The paper concludes with open problems, notably a symmetric chain decomposition for and extensions to other polytopes.

Abstract

We prove that the minimal size of a maximal matching in the permutahedron is asymptotically . On the one hand, we obtain a lower bound by considering -cycles in the permutahedron. On the other hand, we obtain an asymptotical upper bound by multiple applications of Hall's theorem (similar to the approach of Forcade (1973) for the hypercube) and an exact upper bound by an explicit construction. We also derive bounds on minimum maximal matchings in products of permutahedra.

Paper Structure

This paper contains 13 sections, 11 theorems, 21 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. Notation and preliminaries
  3. Matchings.
  4. Permutations.
  5. Permutahedra.
  6. Lower bound for general graphs
  7. Upper bound for bipartite graphs
  8. Explicit construction for permutahedra
  9. Two maximal matchings of $\Pi_4$
  10. Combining maximal matchings
  11. Proof of maximality
  12. Cartesian products of permutahedra
  13. Open questions

Key Result

Theorem 1

The minimal size $\mathcal{M}(\Pi_n)$ of a maximal matching of the permutahedron $\Pi_n$ is bounded by

Figures (2)

  • Figure 1: Matching edges (red) and exposed vertices (blue) for $M^+$ (left) and $M^-$ (right). All other edges are colored according to the exchange position.
  • Figure 2: Matching edges (red) and exposed vertices (blue) for the matching $M$ when ${n = 5}$. All other edges are colored according to the exchange position. The coordinates of the embedding were communicated to us by Nathan Carter. An animated 3d version can be found at https://tinyurl.com/maximalMatchingPermutahedron.

Theorems & Definitions (28)

  • Theorem 1
  • Theorem 2
  • Definition 1
  • Proposition 1
  • proof
  • Example 1
  • Corollary 1
  • proof
  • Remark 1
  • Theorem 3: Hall's theorem MR1581694
  • ...and 18 more