We still need to show that M is a cyclic permutation multigraph, that is, all the edges mentioned above form a red Hamiltonian cycle and a green Hamiltonian cycle over the vertices of M. We start by observing that, within each component G i , these two cycles are symmetric, in the sense that the green Hamiltonian cycle can be obtained from the red Hamiltonian cycle by switching the first and the second indices of each vertex in an edge.
within G i the green Hamiltonian cycle can be obtained from the red Hamiltonian cycle by a rotation along the axis from vertex g 4k,4k .