{
  "id": "2110.09003",
  "version": "v2",
  "published": "2021-10-18T03:55:25.000Z",
  "updated": "2022-10-11T03:20:07.000Z",
  "title": "Application of some techniques in Sperner Theory: Optimal orientations of vertex-multiplications of trees with diameter 4",
  "authors": [
    "W. H. W. Wong",
    "E. G. Tay"
  ],
  "comment": "78 pages",
  "categories": [
    "math.CO"
  ],
  "abstract": "Koh and Tay proved a fundamental classification of $G$ vertex-multiplications into three classes $\\mathscr{C}_0, \\mathscr{C}_1$ and $\\mathscr{C}_2$. They also showed that any vertex-multiplication of a tree with diameter at least 3 does not belong to the class $\\mathscr{C}_2$. Of interest, $G$ vertex-multiplications are extensions of complete $n$-partite graphs and Gutin characterised complete bipartite graphs with an ingenious use of Sperner's Theorem. In this paper, we investigate vertex-multiplications of trees with diameter $4$ in $\\mathscr{C}_0$ (or $\\mathscr{C}_1$) and exhibit its intricate connections with problems in Sperner Theory, thereby extending Gutin's approach. Let $s$ denote the vertex-multiplication of the central vertex. We almost completely characterise the case of even $s$ and give a complete characterisation for the case of odd $s\\ge 3$.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2022-10-11T03:20:07.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C12",
      "05C20",
      "05D05",
      "G.2.2",
      "F.2.2"
    ],
    "keywords": [
      "sperner theory",
      "vertex-multiplication",
      "optimal orientations",
      "gutin characterised complete bipartite graphs",
      "application"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 78,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}