Generalized Gray codes with prescribed ends

Abstract

An $n$-bit Gray code is a sequence of all $n$-bit strings such that consecutive strings differ in a single bit. It is well-known that given $\alpha,\beta\in\{0,1\}^n$, an $n$-bit Gray code between $\alpha$ and $\beta$ exists iff the Hamming distance $d(\alpha,\beta)$ of $\alpha$ and $\beta$ is odd. We generalize this classical result to $k$ pairwise disjoint pairs $\alpha_i, \beta_i\in\{0,1\}^n$: if $d(\alpha_i,\beta_i)$ is odd for all $i$ and $k<n$, then the set of all $n$-bit strings can be partitioned into $k$ sequences such that the $i$-th sequence leads from $\alpha_i$ to $\beta_i$ and consecutive strings 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 $\alpha_i,\beta_i\in\{0,1\}^n$ with $d(\alpha_i,\beta_i)$ odd for which such sequences do not exist.