The minimum forcing number for the torus and hypercube

Let G be a graph with a perfect matching M. Define the forcing number of M in G to be the smallest size of a subset S⊂M that is in no other perfect matching. In this paper, we present a property of bipartite graphs G that acts as a lower bound on the forcing number of perfect matchings in G. We then apply this to the torus and the hypercube, proving that the minimum forcing number of a perfect matching on a 2m×2n torus with m⩾n is 2n, and that the minimum forcing number on an n-dimensional hypercube is 2n/4 if n is even.