{
  "id": "2405.18455",
  "version": "v1",
  "published": "2024-05-28T15:19:56.000Z",
  "updated": "2024-05-28T15:19:56.000Z",
  "title": "Coloring some $(P_6,C_4)$-free graphs with $Δ-1$ colors",
  "authors": [
    "Ran Chen",
    "Di Wu",
    "Xiaowen Zhang"
  ],
  "categories": [
    "math.CO"
  ],
  "abstract": "The Borodin-Kostochka Conjecture states that for a graph $G$, if $\\Delta(G)\\geq9$, then $\\chi(G)\\leq\\max\\{\\Delta(G)-1,\\omega(G)\\}$. We use $P_t$ and $C_t$ to denote a path and a cycle on $t$ vertices, respectively. Let $C=v_1v_2v_3v_4v_5v_1$ be an induced $C_5$. A {\\em $C_5^+$} is a graph obtained from $C$ by adding a $C_3=xyzx$ and a $P_2=t_1t_2$ such that (1) $x$ and $y$ are both exactly adjacent to $v_1,v_2,v_3$ in $V(C)$, $z$ is exactly adjacent to $v_2$ in $V(C)$, $t_1$ is exactly adjacent to $v_4,v_5$ in $V(C)$ and $t_2$ is exactly adjacent to $v_1,v_4,v_5$ in $V(C)$, (2) $t_1$ is exactly adjacent to $z$ in $\\{x,y,z\\}$ and $t_2$ has no neighbors in $\\{x,y,z\\}$. In this paper, we show that the Borodin-Kostochka Conjecture holds for ($P_6,C_4,H$)-free graphs, where $H\\in \\{K_7,C_5^+\\}$. This generalizes some results of Gupta and Pradhan in \\cite{GP21,GP24}.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-05-28T15:19:56.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "free graphs",
      "exactly adjacent",
      "borodin-kostochka conjecture states",
      "borodin-kostochka conjecture holds",
      "generalizes"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}