Generalized Gray codes with prescribed ends of small dimensions

TL;DR

This paper investigates the existence of specific path partitions in hypercubes with prescribed endpoints, providing solutions for small dimensions, advancing the understanding of Gray codes with particular boundary conditions.

Contribution

The paper solves the problem of partitioning hypercube vertices into paths with prescribed odd-distance endpoints for small dimensions, extending known results for Gray codes.

Findings

Established existence of such path partitions for small n

Extended Gray code constructions to new boundary conditions

Provided constructive methods for small hypercube dimensions

Abstract

Given pairwise distinct vertices $\{\alpha_i , \beta_i\}^k_{i=1}$ of the $n$-dimensional hypercube $Q_n$ such that the distance of $\alpha_i$ and $\beta_i$ is odd, are there paths $P_i$ between $\alpha_i$ and $\beta_i$ 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 $n$.