Let x be the vertex of all zeros and y be the vertex of all ones in the hypercube graph ${Q}_{n}$. We have to determine the maximum collection of edge-disjoint xy-paths in ${Q}_{n}$ and a minimum vertex cut of ${Q}_{n}$ separating x and y.

My work: We define a xy-path ${P}_{i}$ as follows:

Start from the vertex x and change the ${i}^{th}$ bit to 1 and then change all bits gradually one by one moving leftward from $i-1$ to 1 to n and back to i.

Note that all paths ${P}_{i}$ are edge-disjoint. Thus we have a maximum collection.

By Menger's Theorem, the second part follows.