{
  "id": "2407.10174",
  "version": "v1",
  "published": "2024-07-14T12:09:25.000Z",
  "updated": "2024-07-14T12:09:25.000Z",
  "title": "On the twin-width of smooth manifolds",
  "authors": [
    "Édouard Bonnet",
    "Kristóf Huszár"
  ],
  "comment": "18 pages, 8 figures",
  "categories": [
    "math.GT",
    "cs.CG",
    "cs.DM"
  ],
  "abstract": "Building on Whitney's classical method of triangulating smooth manifolds, we show that every compact $d$-dimensional smooth manifold admits a triangulation with dual graph of twin-width at most $d^{O(d)}$. In particular, it follows that every compact 3-manifold has a triangulation with dual graph of bounded twin-width. This is in sharp contrast to the case of treewidth, where for any natural number $n$ there exists a closed 3-manifold such that every triangulation thereof has dual graph with treewidth at least $n$. To establish this result, we bound the twin-width of the incidence graph of the $d$-skeleton of the second barycentric subdivision of the $2d$-dimensional hypercubic honeycomb. We also show that every compact, piecewise-linear (hence smooth) $d$-dimensional manifold has triangulations where the dual graph has an arbitrarily large twin-width.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-07-14T12:09:25.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "57Q15",
      "57R05",
      "05C75",
      "57M15",
      "F.2.2",
      "G.2.2"
    ],
    "keywords": [
      "dual graph",
      "dimensional smooth manifold admits",
      "triangulation",
      "dimensional hypercubic honeycomb",
      "second barycentric subdivision"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 18,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}