{
  "id": "1806.09716",
  "version": "v1",
  "published": "2018-06-25T22:19:08.000Z",
  "updated": "2018-06-25T22:19:08.000Z",
  "title": "Stick number of spatial graphs",
  "authors": [
    "Minjung Lee",
    "Sungjong No",
    "Seungsang Oh"
  ],
  "categories": [
    "math.GT"
  ],
  "abstract": "For a nontrivial knot $K$, Negami found an upper bound on the stick number $s(K)$ in terms of its crossing number $c(K)$ which is $s(K) \\leq 2 c(K)$. Later, Huh and Oh utilized the arc index $\\alpha(K)$ to present a more precise upper bound $s(K) \\leq \\frac{3}{2} c(K) + \\frac{3}{2}$. Furthermore, Kim, No and Oh found an upper bound on the equilateral stick number $s_{=}(K)$ as follows; $s_{=}(K) \\leq 2 c(K) +2$. As a sequel to this research program, we similarly define the stick number $s(G)$ and the equilateral stick number $s_{=}(G)$ of a spatial graph $G$, and present their upper bounds as follows; $$ s(G) \\leq \\frac{3}{2} c(G) + 2e + \\frac{3b}{2} -\\frac{v}{2}, $$ $$ s_{=}(G) \\leq 2 c(G) + 2e + 2b - k, $$ where $e$ and $v$ are the number of edges and vertices of $G$, respectively, $b$ is the number of bouquet cut-components, and $k$ is the number of non-splittable components.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-06-25T22:19:08.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "57M25",
      "57M27"
    ],
    "keywords": [
      "spatial graph",
      "equilateral stick number",
      "precise upper bound",
      "arc index",
      "research program"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}