{
  "id": "2402.10633",
  "version": "v1",
  "published": "2024-02-16T12:31:40.000Z",
  "updated": "2024-02-16T12:31:40.000Z",
  "title": "Crossing number of graphs and $\\mathsf{ΔY}$-move",
  "authors": [
    "Youngsik Huh",
    "Ryo Nikkuni"
  ],
  "comment": "17 pages, 14 figures",
  "categories": [
    "math.CO",
    "math.GT"
  ],
  "abstract": "The crossing number of a graph is the minimum number of double points over all generic immersions of the graph into the plane. In this paper we investigate the behavior of crossing number under a graph transformation, called $\\mathsf{\\Delta Y}$-move, on the complete graph $K_n$. Concretely it is shown that for any $k\\in \\mathbb{N}$, there exist a natural number $n$ and a sequence of $\\mathsf{\\Delta Y}$-moves $K_n\\rightarrow G^{(1)}\\rightarrow \\cdots \\rightarrow G^{(k)}$ which is decreasing with respect to the crossing number. We also discuss the decrease of crossing number for relatively small $n$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-02-16T12:31:40.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C62",
      "57M15",
      "05C10"
    ],
    "keywords": [
      "crossing number",
      "minimum number",
      "generic immersions",
      "graph transformation",
      "complete graph"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 17,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}