arXiv subjects
Publications and source records attributed to Václav Koubek.
Given pairwise distinct vertices $\{α_i , β_i\}^k_{i=1}$ of the $n$-dimensional hypercube $Q_n$ such that the distance of $α_i$ and $β_i$ is odd, are there paths $P_i$ between $α_i$ and $β_i$ 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 $n$.
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 $α,β\in\{0,1\}^n$, 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 $α_i, β_i\in\{0,1\}^n$: if $d(α_i,β_i)$ is odd for all $i$ and $k 1$ 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 $α_i,β_i\in\{0,1\}^n$ with $d(α_i,β_i)$ odd for which such sequences do not exist.