Optimal cycles enclosing all the nodes of a $k$-dimensional hypercube

TL;DR

This paper determines the minimal link-length polygonal chain that visits all nodes of a hypercube and constructs such cycles, providing exact solutions for the optimal cycle length in any dimension.

Contribution

It introduces the first exact solution for the minimal link-length cycle visiting all hypercube vertices in any dimension.

Findings

Optimal link-length is 3 * 2^{k-2} for k ≥ 2.

Constructive proof of visiting all nodes with minimal links.

Cycle construction is feasible for all k > 1.

Abstract

We solve the general problem of visiting all the $2^k$ nodes of a $k$-dimensional hypercube by using a polygonal chain that has minimum link-length, and we show that this optimal value is given by $h(2,k):=3 \cdot 2^{k-2}$ if and only if $k \in \mathbb{N}-\{0,1\}$. Furthermore, for any $k$ above one, we constructively prove that it is possible to visit once and only once all the aforementioned nodes, $H(2,k):=\{\{0,1\} \times \{0,1\} \times \dots \times \{0,1\}\} \subset \mathbb{R}^k$, with a cycle (i.e., a closed path) having only $3 \cdot 2^{k-2}$ links.