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}, 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}: if d(αi, βi) is odd for all i and k < n, then the set of all n-bit vectors can be partitioned into k sequences such that the i-th sequence leads from αi to βi and consecutive vectors differ in a single bit. This holds for every n > 1 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} with d(αi, βi) odd for which such sequences do not exist.