{
  "id": "1912.01566",
  "version": "v1",
  "published": "2019-12-03T18:01:30.000Z",
  "updated": "2019-12-03T18:01:30.000Z",
  "title": "On the central levels problem",
  "authors": [
    "Petr Gregor",
    "Ondřej Mička",
    "Torsten Mütze"
  ],
  "categories": [
    "math.CO",
    "cs.DM"
  ],
  "abstract": "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 \\v{S}krekovski, 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.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-12-03T18:01:30.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "dimensional hypercube",
      "hamilton cycle",
      "well-known middle levels problem",
      "central levels problem asserts",
      "classical binary gray codes"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}