{
  "id": "1902.08071",
  "version": "v1",
  "published": "2019-02-21T14:36:15.000Z",
  "updated": "2019-02-21T14:36:15.000Z",
  "title": "Reconfiguration Graph for Vertex Colourings of Weakly Chordal Graphs",
  "authors": [
    "Carl Feghali",
    "Jiří Fiala"
  ],
  "comment": "10 pages, 4 figures",
  "categories": [
    "math.CO",
    "cs.DM"
  ],
  "abstract": "The reconfiguration graph $R_k(G)$ of the $k$-colourings of a graph $G$ contains as its vertex set the $k$-colourings of $G$ and two colourings are joined by an edge if they differ in colour on just one vertex of $G$. We answer a question of Bonamy, Johnson, Lignos, Patel and Paulusma by constructing for each $k \\geq 3$ a $k$-colourable weakly chordal graph $G$ such that $R_{k+1}(G)$ is disconnected. We also introduce a subclass of $k$-colourable weakly chordal graphs which we call $k$-colour compact and show that for each $k$-colour compact graph $G$ on $n$ vertices, $R_{k+1}(G)$ has diameter $O(n^2)$. We show that this class contains all $k$-colourable co-chordal graphs and when $k = 3$ all $3$-colourable $(P_5, \\overline{P_5}, C_5)$-free graphs. We also mention some open problems.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-02-21T14:36:15.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C15"
    ],
    "keywords": [
      "reconfiguration graph",
      "vertex colourings",
      "colourable weakly chordal graph",
      "colour compact graph",
      "open problems"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 10,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}