{
  "id": "1910.00007",
  "version": "v1",
  "published": "2019-09-30T06:28:36.000Z",
  "updated": "2019-09-30T06:28:36.000Z",
  "title": "On a conjecture of Badakhshian, Katona and Tuza",
  "authors": [
    "Yeshwant Pandit",
    "S. L. Sravanthi",
    "Suresh Dara",
    "S. M. Hegde"
  ],
  "categories": [
    "math.CO"
  ],
  "abstract": "Let ${[n] \\choose k}$ and ${[n] \\choose l}$ $( k > l ) $ where $[n] = \\{1,2,3,...,n\\}$ denote the family of all $k$-element subsets and $l$-element subsets of $[n]$ respectively. Define a bipartite graph $G_{k,l} = ({[n] \\choose k},{[n] \\choose l},E)$ such that two vertices $S\\, \\epsilon \\,{[n] \\choose k} $ and $T\\, \\epsilon \\,{[n] \\choose l} $ are adjacent if and only if $T \\subset S$. In this paper, we disprove the conjecture of Badakhshian, Katona and Tuza for $k= \\lceil \\frac{n}{2} \\rceil +1 $ and $k=n-1$ for $n\\geq 10$ and $n>2$ respectively. Also, we give an upper bound for the domination number of $G_{k,2}$ for $k > \\lceil \\frac{n}{2} \\rceil$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-09-30T06:28:36.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "conjecture",
      "badakhshian",
      "element subsets",
      "bipartite graph",
      "upper bound"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}