Minimum maximal matchings in broader classes of polytopes

Determine upper and lower bounds for the size of a minimum maximal matching in broader classes of polytopes generalizing the permutahedron and associahedron, including quotientopes, bipartite or regular quotientopes, graphical zonotopes, graph associahedra, and Coxeter permutahedra.

Background

The paper develops bounds for minimum maximal matchings in permutahedra and extends them to Cartesian products of permutahedra, which form a class of bipartite regular quotientopes. It also discusses how the general lower-bound method applies to associahedra and Coxeter permutahedra, while noting limitations for graphical zonotopes.

The authors leave unresolved whether comparable upper and lower bounds can be obtained across larger and structurally related classes of polytopes. The question concerns, in particular, quotientopes and several related families, even under restrictions such as bipartiteness or regularity.