{
  "id": "2203.06594",
  "version": "v1",
  "published": "2022-03-13T07:53:26.000Z",
  "updated": "2022-03-13T07:53:26.000Z",
  "title": "On join product and local antimagic chromatic number of regular 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 if $G$ admits a local antimagic labeling. A bijection $f : E \\to \\{1,2,\\ldots,q\\}$ is called a local antimagic labeling of $G$ if for any two adjacent vertices $u$ and $v$, we have $f^+(u) \\ne f^+(v)$, where $f^+(u) = \\sum_{e\\in E(u)} f(e)$, and $E(u)$ is the set of edges incident to $u$. Thus, any local antimagic labeling induces a proper vertex coloring of $G$ if vertex $v$ is assigned the color $f^+(v)$. The local antimagic chromatic number, denoted $\\chi_{la}(G)$, is the minimum number of induced colors taken over local antimagic labeling of $G$. Let $G$ and $H$ be two vertex disjoint graphs. The join graph of $G$ and $H$, denoted $G \\vee H$, is the graph $V(G\\vee H) = V(G) \\cup V(H)$ and $E(G\\vee H) = E(G) \\cup E(H) \\cup \\{uv \\,|\\, u\\in V(G), v \\in V(H)\\}$. In this paper, we show the existence of non-complete regular graphs with arbitrarily large order, regularity and local antimagic chromatic numbers.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-03-13T07:53:26.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C78",
      "05C69"
    ],
    "keywords": [
      "local antimagic chromatic number",
      "join product",
      "local antimagic labeling induces",
      "vertex disjoint graphs",
      "non-complete regular graphs"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}