Generalized Gray codes with prescribed ends of small dimensions

Abstract: Given pairwise distinct vertices ${\alpha_i , \beta_i}^k_{i=1}$ of the $n$-dimensional hypercube $Q_n$ such that the distance of $\alpha_i$ and $\beta_i$ is odd, are there paths $P_i$ between $\alpha_i$ and $\beta_i$ such that ${V (P_i)}^k_{i=1}$ partitions $V(Q_n)$? A positive solution for every $n\ge1$ and $k=1$ is known as a Gray code of dimension $n$. In this paper we settle this problem for small values of $n$.