{
  "id": "2107.01033",
  "version": "v1",
  "published": "2021-07-01T06:24:22.000Z",
  "updated": "2021-07-01T06:24:22.000Z",
  "title": "$L(n)$ graphs are vertex-pancyclic and Hamilton-connected",
  "authors": [
    "S. Morteza Mirafzal",
    "Sara Kouhi"
  ],
  "comment": "7 pages. 1 figure",
  "categories": [
    "math.CO"
  ],
  "abstract": "A graph $G$ of order $n>2$ is pancyclic if $G$ contains a cycle of length $l$ for each integer $l$ with $3 \\leq l \\leq n $ and it is called vertex-pancyclic if every vertex is contained in a cycle of length $l$ for every $3 \\leq l \\leq n $. A graph $G$ of order $n > 2$ is Hamilton-connected if for any pair of distinct vertices $u$ and $v$, there is a Hamilton $u$-$v$ path, namely, there is a $u$-$v$ path of length $n-1$. The graph $ B(n)$ is a graph with the vertex set $V=\\{v \\ | \\ v \\subset [n] , | v | \\in \\{ 1,2 \\} \\} $ and the edge set $ E= \\{ \\{ v , w \\} \\ | \\ v , w \\in V , v \\subset w $ or $ w \\subset v \\}$, where $[n]=\\{1,2,...,n\\}$. We denote by $L(n)$ the line graph of $B(n)$, that is, $L(n)=L(B(n))$. In this paper, we show that the graph $L(n)$ is vertex-pancyclic and Hamilton-connected whenever $n\\geq 6$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-07-01T06:24:22.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C38",
      "90B10"
    ],
    "keywords": [
      "vertex-pancyclic",
      "edge set",
      "distinct vertices",
      "vertex set",
      "line graph"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 7,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}