{
  "id": "2007.00115",
  "version": "v1",
  "published": "2020-06-30T21:23:11.000Z",
  "updated": "2020-06-30T21:23:11.000Z",
  "title": "Hamiltonicity of the Double Vertex Graph and the Complete Double Vertex Graph of some Join Graphs",
  "authors": [
    "Luis Adame",
    "Luis Manuel Rivera",
    "Ana Laura Trujillo-Negrete"
  ],
  "categories": [
    "math.CO"
  ],
  "abstract": "Let $G$ be a simple graph of order $n$. The double vertex graph $F_2(G)$ of $G$ is the graph whose vertices are the $2$-subsets of $V(G)$, where two vertices are adjacent in $F_2(G)$ if their symmetric difference is a pair of adjacent vertices in $G$. A generalization of this graph is the complete double vertex graph $M_2(G)$ of $G$, defined as the graph whose vertices are the $2$-multisubsets of $V(G)$, and two of such vertices are adjacent in $M_2(G)$ if their symmetric difference (as multisets) is a pair of adjacent vertices in $G$. In this paper we exhibit an infinite family of graphs (containing Hamiltonian and non-Hamiltonian graphs) for which $F_2(G)$ and $M_2(G)$ are Hamiltonian. This family of graphs is the set of join graphs $G=G_1 + G_2$, where $G_1$ and $G_2$ are of order $m\\geq 1$ and $n\\geq 2$, respectively, and $G_2$ has a Hamiltonian path. For this family of graphs, we show that if $m\\leq 2n$ then $F_2(G)$ is Hamiltonian, and if $m\\leq 2(n-1)$ then $M_2(G)$ is Hamiltonian.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-06-30T21:23:11.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C45",
      "05C76"
    ],
    "keywords": [
      "complete double vertex graph",
      "join graphs",
      "symmetric difference",
      "adjacent vertices",
      "hamiltonicity"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}