Long cycles in the middle two layers of the discrete cube

We prove that the middle two layers of the cube Q2r+1 contain a cycle of length (1−o(1))22r+1r. Our methods can also be used to show that the odd graph Ok contains a cycle of length (1−o(1))|V(Ok)|. our tools, giving a Hamilton cycle in the cube with the minimum number of ‘changes of direction’, may be of independent interest.