The -analog of the middle levels problem

The well-known middle levels problem is to find a Hamiltonian cycle in the graph induced from the binary Hamming graph H 2 ( 2 k + 1 ) by the words of weight k or k + 1 . In this paper we define the q -analog of the middle levels problem. Let n = 2 k + 1 and let  q  be a power of a prime number. Consider the set of ( k + 1 ) -dimensional subspaces and the set of k -dimensional subspaces of F q n . Can these subspaces be ordered in a way that for any two adjacent subspaces X and Y , either X ⊂ Y or Y ⊂ X ? A construction method which yields many Hamiltonian cycles for any given q and k = 2 is presented.