• DOI:10.4230/LIPIcs.ICALP.2020.60
• Corpus ID: 208548372

On the central levels problem

@article{Gregor2019OnTC,
  title={On the central levels problem},
  author={Petr Gregor and Ondrej Micka and Torsten M{\"u}tze},
  journal={ArXiv},
  year={2019},
  volume={abs/1912.01566}
}

• Petr Gregor, O. Micka, Torsten Mütze
• Published 3 December 2019
• Computer Science
• ArXiv

The central levels problem asserts that the subgraph of the $(2m+1)$-dimensional hypercube induced by all bitstrings with at least $m+1-\ell$ many 1s and at most $m+\ell$ many 1s, i.e., the vertices in the middle $2\ell$ levels, has a Hamilton cycle for any $m\geq 1$ and $1\le \ell\le m+1$. This problem was raised independently by Savage, by Gregor and Skrekovski, and by Shen and Williams, and it is a common generalization of the well-known middle levels problem, namely the case $\ell=1$, and classical binary Gray codes, namely the case $\ell=m+1$. In this paper we present a general constructive solution of the central levels problem. Our results also imply the existence of optimal cycles through any sequence of $\ell$ consecutive levels in the $n$-dimensional hypercube for any $n\ge 1$ and $1\le \ell \le n+1$. Moreover, extending an earlier construction by Streib and Trotter, we construct a Hamilton cycle through the $n$-dimensional hypercube, $n\geq 2$, that contains the symmetric chain decomposition constructed by Greene and Kleitman in the 1970s, and we provide a loopless algorithm for computing the corresponding Gray code.