Dominants and submissives of matching polyhedra

Abstract

LetP be the convex hull of perfect matchings of a graphG=(V, E). The dominant ofP is {y∈R ^E∶y≥x for somex∈P}. A theorem of Fulkerson implies that, ifG is bipartite, then the dominant ofP can be described by linear inequalities having {0, 1}-valued coefficients. However, this is far from true in general. Here it is proved that, for every positive integern, there exists a graph for which the dominant has an essential valid inequality whose coefficient-set includes the firstn positive integers. A similar result holds for the submissive ofP, {y∈R ^E∶0≤y≤x for somex∈P}.