Generalized Gray codes with prescribed ends of small dimensions

Given pairwise distinct vertices {α_i , β_i}^k_i=1 of the n-dimensional hypercube Q_n such that the distance of α_i and β_i is odd, are there paths P_i between α_i and β_i such that {V (P_i)}^k_i=1 partitions V(Q_n)? A positive solution for every n≥1 and k=1 is known as a Gray code of dimension n. In this paper we settle this problem for small values of n.