{
  "id": "1901.02468",
  "version": "v1",
  "published": "2019-01-08T19:00:11.000Z",
  "updated": "2019-01-08T19:00:11.000Z",
  "title": "Schur and $e$-positivity of trees and cut vertices",
  "authors": [
    "Samantha Dahlberg",
    "Adrian She",
    "Stephanie van Willigenburg"
  ],
  "comment": "20 pages",
  "categories": [
    "math.CO"
  ],
  "abstract": "We prove that the chromatic symmetric function of any $n$-vertex tree containing a vertex of degree $d\\geq \\log _2n +1$ is not $e$-positive, that is, not a positive linear combination of elementary symmetric functions. Generalizing this, we also prove that the chromatic symmetric function of any $n$-vertex graph containing a cut vertex whose deletion disconnects the graph into $d\\geq\\log _2n +1$ connected components is not $e$-positive. Furthermore we prove that any $n$-vertex bipartite graph, including all trees, containing a vertex of degree greater than $\\lceil \\frac{n}{2}\\rceil$ is not Schur-positive, namely not a positive linear combination of Schur functions.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-01-08T19:00:11.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05E05",
      "05C05",
      "05C15",
      "05C70",
      "16T30",
      "20C30"
    ],
    "keywords": [
      "cut vertex",
      "chromatic symmetric function",
      "positive linear combination",
      "positivity",
      "vertex bipartite graph"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 20,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}