On generalized middle-level problem

Let image be the subgraph of the hypercube image induced by levels between k and image, where image is odd. The well-known middle-level conjecture asserts that image is Hamiltonian for all image. We study this problem in image for fixed k. It is known that image and image are Hamiltonian for all odd image. In this paper we prove that also image is Hamiltonian for all odd image, and we conjecture that image is Hamiltonian for every image and every odd image.