 Flexible vertex in a balanced bipartite graph

A vertex on one side of a balanced bipartite graph is flexible if every incident edge lies in a perfect matching. The strict Hall marriage theorem inequalities for nonempty proper subsets imply flexibility of every vertex: after forcing an edge, removal of its ends loses at most one neighbour from each remaining set. If a perfect matching exists without the strict condition, choose a minimal nonempty tight set. Its induced balanced graph has the strict condition internally, and each internal matching extends using the original matching outside that set. Thus at least one flexible vertex always exists.