{
  "id": "1810.07719",
  "version": "v1",
  "published": "2018-10-17T18:12:38.000Z",
  "updated": "2018-10-17T18:12:38.000Z",
  "title": "Further Results on Existentially Closed Graphs Arising from Block Designs",
  "authors": [
    "Xiao-Nan Lu"
  ],
  "comment": "10 pages",
  "categories": [
    "math.CO"
  ],
  "abstract": "A graph is $n$-existentially closed ($n$-e.c.) if for any disjoint vertex $A$, $B$ of vertices with $|{A \\cup B}|=n$, there is a vertex $z \\notin A \\cup B$ adjacent to every vertex of $A$ and no vertex of $B$. For a block design with block set $\\mathcal{B}$, its block intersection graph is the graph whose vertex set is $\\mathcal{B}$ and two vertices (blocks) are adjacent if they have non-empty intersection. In this paper, we investigate the block intersection graphs of pairwise balanced designs, and propose a sufficient condition for such graphs to be $2$-e.c. In particular, we study the $\\lambda$-fold triple systems with $\\lambda \\ge 2$ and determine for which parameters their block intersection graphs are $1$- or $2$-e.c. Moreover, for Steiner quadruple systems, the block intersection graphs and their analogue called $\\{1\\}$-block intersection graphs are investigated, and the necessary and sufficient conditions when such graphs are $2$-e.c. are established.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-10-17T18:12:38.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05B07",
      "05B05",
      "05C75"
    ],
    "keywords": [
      "block intersection graph",
      "existentially closed graphs arising",
      "block design",
      "sufficient condition",
      "steiner quadruple systems"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 10,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}