{
  "id": "2209.06171",
  "version": "v1",
  "published": "2022-09-13T17:20:25.000Z",
  "updated": "2022-09-13T17:20:25.000Z",
  "title": "Proving a directed analogue of the Gyárfás-Sumner conjecture for orientations of $P_4$",
  "authors": [
    "Linda Cook",
    "Tomáš Masařík",
    "Marcin Pilipczuk",
    "Amadeus Reinald",
    "Uéverton S. Souza"
  ],
  "comment": "24 pages, 5 figures",
  "categories": [
    "math.CO",
    "cs.DM"
  ],
  "abstract": "An oriented graph is a digraph that does not contain a directed cycle of length two. An (oriented) graph $D$ is $H$-free if $D$ does not contain $H$ as an induced sub(di)graph. The Gy\\'arf\\'as-Sumner conjecture is a widely-open conjecture on simple graphs, which states that for any forest $F$, there is some function $f$ such that every $F$-free graph $G$ with clique number $\\omega(G)$ has chromatic number at most $f(\\omega(G))$. Aboulker, Charbit, and Naserasr [Extension of Gy\\'arf\\'as-Sumner Conjecture to Digraphs; E-JC 2021] proposed an analogue of this conjecture to the dichromatic number of oriented graphs. The dichromatic number of a digraph $D$ is the minimum number of colors required to color the vertex set of $D$ so that no directed cycle in $D$ is monochromatic. Aboulker, Charbit, and Naserasr's $\\overrightarrow{\\chi}$-boundedness conjecture states that for every oriented forest $F$, there is some function $f$ such that every $F$-free oriented graph $D$ has dichromatic number at most $f(\\omega(D))$, where $\\omega(D)$ is the size of a maximum clique in the graph underlying $D$. In this paper, we perform the first step towards proving Aboulker, Charbit, and Naserasr's $\\overrightarrow{\\chi}$-boundedness conjecture by showing that it holds when $F$ is any orientation of a path on four vertices.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-09-13T17:20:25.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C15",
      "05C20",
      "05C38",
      "05C69"
    ],
    "keywords": [
      "gyárfás-sumner conjecture",
      "directed analogue",
      "dichromatic number",
      "oriented graph",
      "orientation"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 24,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}