{
  "id": "2206.03422",
  "version": "v1",
  "published": "2022-06-07T16:15:24.000Z",
  "updated": "2022-06-07T16:15:24.000Z",
  "title": "Vertex-critical $(P_3+\\ell P_1)$-free and vertex-critical (gem, co-gem)-free graphs",
  "authors": [
    "Tala Abuadas",
    "Ben Cameron",
    "Chính T. Hoàng",
    "Joe Sawada"
  ],
  "categories": [
    "math.CO",
    "cs.DM"
  ],
  "abstract": "A graph $G$ is $k$-vertex-critical if $\\chi(G)=k$ but $\\chi(G-v)<k$ for all $v\\in V(G)$ where $\\chi(G)$ denotes the chromatic number of $G$. We show that there are only finitely many $k$-critical $(P_3+\\ell P_1)$-free graphs for all $k$ and all $\\ell$. Together with previous results, the only graphs $H$ for which it is unknown if there are an infinite number of $k$-vertex-critical $H$-free graphs is $H=(P_4+\\ell P_1)$ for all $\\ell\\ge 1$. We consider a restriction on the smallest open case, and show that there are only finitely many $k$-vertex-critical (gem, co-gem)-free graphs for all $k$, where gem$=\\overline{P_4+P_1}$. To do this, we show the stronger result that every vertex-critical (gem, co-gem)-free graph is either complete or a clique expansion of $C_5$. This characterization allows us to give the complete list of all $k$-vertex-critical (gem, co-gem)-free graphs for all $k\\le 16$",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-06-07T16:15:24.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "vertex-critical",
      "free graphs",
      "smallest open case",
      "chromatic number",
      "infinite number"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}