{
  "id": "2501.05909",
  "version": "v1",
  "published": "2025-01-10T12:10:11.000Z",
  "updated": "2025-01-10T12:10:11.000Z",
  "title": "2-extendability of (4,5,6)-fullerenes",
  "authors": [
    "Lifang Zhao",
    "Heping Zhang"
  ],
  "comment": "28 pages,15 figures",
  "categories": [
    "math.CO"
  ],
  "abstract": "A (4,5,6)-fullerene is a plane cubic graph whose faces are only quadrilaterals, pentagons and hexagons, which includes all (4,6)- and (5,6)-fullerenes. A connected graph $G$ with at least $2k+2$ vertices is $k$-extendable if $G$ has perfect matchings and any matching of size $k$ is contained in a perfect matching of $G$. We know that each (4,5,6)-fullerene graph is 1-extendable and at most 2-extendable. It is natural to wonder which (4,5,6)-fullerene graphs are 2-extendable. In this paper, we completely solve this problem (see Theorem 3.3): All non-2-extendable (4,5,6)-fullerenes consist of four sporadic (4,5,6)-fullerenes ($F_{12},F_{14},F_{18}$ and $F_{20}$) and five classes of (4,5,6)-fullerenes. As a surprising consequence, we find that all (4,5,6)-fullerenes with the anti-Kekul\\'{e} number 3 are non-2-extendable. Further, there also always exists a non-2-extendable (4,5,6)-fullerene with arbitrarily even $n\\geqslant10$ vertices.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2025-01-10T12:10:11.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "G.2.2"
    ],
    "keywords": [
      "plane cubic graph",
      "perfect matching",
      "connected graph",
      "quadrilaterals",
      "surprising consequence"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 28,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}