Optimal cycles enclosing all the nodes of a $k$-dimensional hypercube
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.
- Publication:
-
arXiv e-prints
- Pub Date:
- December 2022
- DOI:
- 10.48550/arXiv.2212.11216
- arXiv:
- arXiv:2212.11216
- Bibcode:
- 2022arXiv221211216R
- Keywords:
-
- Mathematics - Combinatorics;
- 05C38 (Primary) 05C12;
- 91A43 (Secondary)
- E-Print:
- 13 pages, 5 figures. Typos corrected