{
  "id": "2405.01890",
  "version": "v1",
  "published": "2024-05-03T07:26:50.000Z",
  "updated": "2024-05-03T07:26:50.000Z",
  "title": "Counterexamples to two conjectures on mean color numbers of graphs",
  "authors": [
    "Wushuang Zhai",
    "Yan Yang"
  ],
  "comment": "5 pages",
  "categories": [
    "math.CO"
  ],
  "abstract": "The mean color number of an $n$-vertex graph $G$, denoted by $\\mu(G)$, is the average number of colors used in all proper $n$-colorings of $G$. For any graph $G$ and a vertex $w$ in $G$, Dong (2003) conjectured that if $H$ is a graph obtained from a graph $G$ by deleting all but one of the edges which are incident to $w$, then $\\mu(G)\\geq \\mu(H)$; and also conjectured that $\\mu(G)\\geq \\mu((G-w)\\cup K_1)$. We prove that there is an infinite family of counterexamples to these two conjectures.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-05-03T07:26:50.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C15",
      "05C30",
      "05C31"
    ],
    "keywords": [
      "mean color number",
      "counterexamples",
      "conjectures",
      "vertex graph"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 5,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}