Generalized Gray codes with prescribed ends

TL;DR

This paper extends the classical Gray code existence result to multiple disjoint pairs with prescribed start and end points, identifying conditions under which such sequences exist or do not exist.

Contribution

It generalizes the classical Gray code result to multiple pairs, providing necessary and sufficient conditions for their existence with a specific exception.

Findings

Sequences exist for all cases with odd Hamming distances except one

The result is optimal; some configurations do not admit such sequences

Identifies an exception when n=4 and k=3

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.