Abstract: The perfect matching polytope of a graph G is the convex hull of the set of incidence vectors of its perfect matchings. Edmonds (1965) gave an exact characterization of the perfect matching polytope; in particular, he showed that a vector belongs to this polytope if and only if it satisfies (1) non-negativity constraints, (2) degree constraints and (3) odd set constraints. Carvalho, Lucchesi and Murty (2004) characterized graphs whose perfect matching polytopes are determined by non-negativity and the degree constraints. It is well-known that bipartite graphs have this property. I will discuss their result and some related work.
Reference: http://www.sciencedirect.com/science/article/pii/S0095895604000772