{
  "id": "1811.04902",
  "version": "v1",
  "published": "2018-11-12T18:41:49.000Z",
  "updated": "2018-11-12T18:41:49.000Z",
  "title": "The Erd\\H{os}-Ko-Rado property of trees of depth two",
  "authors": [
    "Carl Feghali"
  ],
  "comment": "10 pages, 1 figure",
  "categories": [
    "math.CO"
  ],
  "abstract": "A family of sets is intersecting if any two sets in the family intersect. Given a graph $G$ and an integer $r\\geq 1$, let $\\mathcal{I}^{(r)}(G)$ denote the family of independent sets of size $r$ of $G$. For a vertex $v$ of $G$, let $\\mathcal{I}^{(r)}_v(G)$ denote the family of independent sets of size $r$ that contain $v$. This family is called an $r$-star. Then $G$ is said to be $r$-EKR if no intersecting subfamily of $ \\mathcal{I}^{(r)}(G)$ is bigger than the largest $r$-star. Let $n, r \\geq 1$, $k \\geq 2$, and let $T(n, k)$ be the tree of depth two in which the root has degree $n$ and every neighbour of the root has the same number $k + 1$ of neighbours. For each $k \\geq 2$, we show that $T(n, k)$ is $r$-EKR if $2r \\leq n$, extending results of Borg and of Feghali, Johnson and Thomas who considered the case $k = 1$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-11-12T18:41:49.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05D05"
    ],
    "keywords": [
      "independent sets",
      "family intersect",
      "extending results"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 10,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}