{
  "id": "2401.14653",
  "version": "v1",
  "published": "2024-01-26T05:24:54.000Z",
  "updated": "2024-01-26T05:24:54.000Z",
  "title": "Complete characterization of graphs with local total antimagic chromatic number 3",
  "authors": [
    "G. C. Lau"
  ],
  "comment": "18 pages, 8 figures",
  "categories": [
    "math.CO"
  ],
  "abstract": "A total labeling of a graph $G = (V, E)$ is said to be local total antimagic if it is a bijection $f: V\\cup E \\to\\{1,\\ldots ,|V|+|E|\\}$ such that adjacent vertices, adjacent edges, and incident vertex and edge have distinct induced weights where the induced weight of a vertex $v$, $w_f(v) = \\sum f(e)$ with $e$ ranging over all the edges incident to $v$, and the induced weight of an edge $uv$ is $w_f(uv) = f(u) + f(v)$. The local total antimagic chromatic number of $G$, denoted by $\\chi_{lt}(G)$, is the minimum number of distinct induced vertex and edge weights over all local total antimagic labelings of $G$. In this paper, we first obtained general lower and upper bounds for $\\chi_{lt}(G)$ and sufficient conditions to construct a graph $H$ with $k$ pendant edges and $\\chi_{lt}(H) \\in\\{\\Delta(H)+1, k+1\\}$. We then completely characterized graphs $G$ with $\\chi_{lt}(G)=3$. Many families of (disconnected) graphs $H$ with $k$ pendant edges and $\\chi_{lt}(H) \\in\\{\\Delta(H)+1, k+1\\}$ are also obtained.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-01-26T05:24:54.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C78",
      "05C69"
    ],
    "keywords": [
      "local total antimagic chromatic number",
      "complete characterization",
      "induced weight",
      "pendant edges",
      "local total antimagic labelings"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 18,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}