Minimal maximal matchings in general Cartesian products

Characterize minimal maximal matchings in Cartesian products of graphs in general, including their minimum possible cardinalities and corresponding upper and lower bounds.

Background

The paper proves bounds for Cartesian products of permutahedra and obtains an exact one-third upper bound in the non-hypercube cases considered. Its construction relies on special maximal matchings in permutahedra and on bipartite-factor structure.

The authors explicitly leave unresolved what can be said for Cartesian products beyond this specialized setting. A general theory would identify how the minimum maximal matching number behaves under Cartesian products and which structural properties of the factors yield tight bounds or constructions.