Special parity of perfect matchings in bipartite graphs

Abstract Let G be a bipartite graph in which every edge belongs to some perfect matching, and let D be a subset of its edge set. It is shown that M ∩ D has the same parity for every perfect matching M if and only if D is a cut, and equivalently if and only if ( G, D ) is a balanced signed-graph. This gives necessary and sufficient conditions on the sign pattern of an n × n real matrix under which all nonzero terms in its permanent expansion have the same sign.