{
  "id": "2008.09754",
  "version": "v1",
  "published": "2020-08-22T04:26:30.000Z",
  "updated": "2020-08-22T04:26:30.000Z",
  "title": "On Local Antimagic Chromatic Number of Spider Graphs",
  "authors": [
    "Gee-Choon Lau",
    "Wai-Chee Shiu",
    "Chee-Xian Soo"
  ],
  "comment": "25 pages",
  "categories": [
    "math.CO"
  ],
  "abstract": "An edge labeling of a connected graph $G = (V,E)$ is said to be local antimagic if it is a bijection $f : E \\to \\{1, . . . , |E|\\}$ such that for any pair of adjacent vertices $x$ and $y$, $f^+(x) \\ne f^+(y)$, where the induced vertex label $f^+(x) = \\sum f(e)$, with $e$ ranging over all the edges incident to $x$. The local antimagic chromatic number of $G$, denoted by $\\chi_{la}(G)$, is the minimum number of distinct induced vertex labels over all local antimagic labelings of $G$. In this paper, we first show that a $d$-leg spider graph has $d+1\\le \\chi_{la}\\le d+2$. We then obtain many sufficient conditions such that both the values are attainable. Finally, we show that each 3-leg spider has $\\chi_{la} = 4$ if not all legs are of odd length. We conjecture that almost all $d$-leg spiders of size $q$ that satisfies $d(d+1) \\le 2(2q-1)$ with each leg length at least 2 has $\\chi_{la} = d+1$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-08-22T04:26:30.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C78",
      "05C69"
    ],
    "keywords": [
      "local antimagic chromatic number",
      "leg spider graph",
      "distinct induced vertex labels",
      "local antimagic labelings",
      "edges incident"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 25,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}