Generalized Gray codes with prescribed ends

An n-bit Gray code is a sequence of all n-bit vectors such that consecutive vectors differ in a single bit. It is well-known that given α,β∈{0,1}n, an n-bit Gray code between α and β exists iff the Hamming distance d(α,β) of α and β is odd. We generalize this classical result to k pairwise disjoint pairs αi,βi∈{0,1}n: if d(αi,βi) is odd for all i and k1 with one exception in the case when n=k+1=4. Our result is optimal in the sense that for every n>2 there are n pairwise disjoint pairs αi,βi∈{0,1}n with d(αi,βi) odd for which such sequences do not exist.