{
  "id": "2208.14707",
  "version": "v1",
  "published": "2022-08-31T09:13:32.000Z",
  "updated": "2022-08-31T09:13:32.000Z",
  "title": "A note on local antimagic chromatic number of lexicographic product graphs",
  "authors": [
    "Gee-Choon Lau",
    "Wai-Chee Shiu",
    "K. Premalatha",
    "Ruixue Zhang",
    "M. Nalliah"
  ],
  "categories": [
    "math.CO"
  ],
  "abstract": "Let $G = (V,E)$ be a connected simple graph. A bijection $f: E \\rightarrow \\{1,2,\\ldots,|E|\\}$ is called a local antimagic labeling of $G$ if $f^+(u) \\neq f^+(v)$ holds for any two adjacent vertices $u$ and $v$, where $f^+(u) = \\sum_{e\\in E(u)} f(e)$ and $E(u$) is the set of edges incident to $u$. A graph $G$ is called local antimagic if $G$ admits at least a local antimagic labeling. The local antimagic chromatic number, denoted $\\chi_{la}(G)$, is the minimum number of induced colors taken over local antimagic labelings of $G$. Let $G$ and $H$ be two disjoint graphs. The graph $G[H]$ is obtained by the lexicographic product of $G$ and $H$. In this paper, we obtain sufficient conditions for $\\chi_{la}(G[H])\\leq \\chi_{la}(G)\\chi_{la}(H)$. Consequently, we give examples of $G$ and $H$ such that $\\chi_{la}(G[H]) = \\chi(G)\\chi(H)$, where $\\chi(G)$ is the chromatic number of $G$. We conjecture that (i) there are infinitely many graphs $G$ and $H$ such that $\\chi_{la}(G[H])=\\chi_{la}(G)\\chi_{la}(H) = \\chi(G)\\chi(H)$, and (ii) for $k\\ge 1$, $\\chi_{la}(G[H]) = \\chi(G)\\chi(H)$ if and only if $\\chi(G)\\chi(H) = 2\\chi(H) + \\lceil\\frac{\\chi(H)}{k}\\rceil$, where $2k+1$ is the length of a shortest odd cycle in $G$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-08-31T09:13:32.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "local antimagic chromatic number",
      "lexicographic product graphs",
      "local antimagic labeling",
      "shortest odd cycle",
      "adjacent vertices"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}