{
  "id": "2203.06337",
  "version": "v1",
  "published": "2022-03-12T04:09:22.000Z",
  "updated": "2022-03-12T04:09:22.000Z",
  "title": "On local antimagic total labeling of amalgamation graphs",
  "authors": [
    "Gee-Choon Lau",
    "Wai-Chee Shiu"
  ],
  "categories": [
    "math.CO"
  ],
  "abstract": "Let $G = (V,E)$ be a connected simple graph of order $p$ and size $q$. A graph $G$ is called local antimagic (total) if $G$ admits a local antimagic (total) labeling. A bijection $g : E \\to \\{1,2,\\ldots,q\\}$ is called a local antimagic labeling of $G$ if for any two adjacent vertices $u$ and $v$, we have $g^+(u) \\ne g^+(v)$, where $g^+(u) = \\sum_{e\\in E(u)} g(e)$, and $E(u)$ is the set of edges incident to $u$. Similarly, a bijection $f:V(G)\\cup E(G)\\to \\{1,2,\\ldots,p+q\\}$ is called a local antimagic total labeling of $G$ if for any two adjacent vertices $u$ and $v$, we have $w_f(u)\\ne w_f(v)$, where $w_f(u) = f(u) + \\sum_{e\\in E(u)} f(e)$. Thus, any local antimagic (total) labeling induces a proper vertex coloring of $G$ if vertex $v$ is assigned the color $g^+(v)$ (respectively, $w_f(u)$). The local antimagic (total) chromatic number, denoted $\\chi_{la}(G)$ (respectively $\\chi_{lat}(G)$), is the minimum number of induced colors taken over local antimagic (total) labeling of $G$. In this paper, we determined $\\chi_{lat}(G)$ where $G$ is the amalgamation of complete graphs.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-03-12T04:09:22.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C78",
      "05C15"
    ],
    "keywords": [
      "local antimagic total labeling",
      "amalgamation graphs",
      "adjacent vertices",
      "complete graphs",
      "minimum number"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}