The q-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(2k + 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 = 2k + 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 subset of Y or Y subset of X? A construction method which yields many Hamiltonian cycles for any given q and k = 2 is presented. (C) 2014 Elsevier B.V. All rights reserved.