{
  "id": "2001.05138",
  "version": "v1",
  "published": "2020-01-15T05:39:52.000Z",
  "updated": "2020-01-15T05:39:52.000Z",
  "title": "On number of pendants in local antimagic chromatic number",
  "authors": [
    "Gee-Choon Lau",
    "Wai-Chee Shiu",
    "Ho-Kuen Ng"
  ],
  "comment": "6 page, 3 figures, a new short paper that gives tight upper and lower bounds with sufficient conditions for the bounds to be equal",
  "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 $f:E \\to\\{1,\\ldots ,|E|\\}$ such that for any pair of adjacent vertices $x$ and $y$, $f^+(x)\\not= f^+(y)$, 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 $G$. Let $\\chi(G)$ be the chromatic number of $G$. In this paper, sharp upper and lower bounds of $\\chi_{la}(G)$ for $G$ with pendant vertices, and sufficient conditions for the bounds to equal, are obtained. Consequently, for $k\\ge 1$, there are infinitely many graphs with $k \\ge \\chi(G) - 1$ pendant vertices and $\\chi_{la}(G) = k+1$. We conjecture that every tree $T_k$, other than certain caterpillars, spiders and lobsters, with $k\\ge 1$ pendant vertices has $\\chi_{la}(T_k) = k+1$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-01-15T05:39:52.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C78",
      "05C69"
    ],
    "keywords": [
      "local antimagic chromatic number",
      "pendant vertices",
      "distinct induced vertex labels",
      "local antimagic labelings",
      "minimum number"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 6,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}