{
  "id": "1906.10332",
  "version": "v1",
  "published": "2019-06-25T05:57:28.000Z",
  "updated": "2019-06-25T05:57:28.000Z",
  "title": "Every Graph Is Local Antimagic Total",
  "authors": [
    "Gee-Choon Lau"
  ],
  "comment": "4 pages",
  "categories": [
    "math.CO"
  ],
  "abstract": "An antimagic labelling of a graph $G$ is a bijection $h : E(G) \\to \\{1, \\ldots, |E(G)|\\}$ such that the induced vertex label $h^+(v) = \\sum_{uv\\in E(G)}$ distinguish all vertices $v$. A well-known conjecture of Hartsfield and Ringel (1994) is that every connected graph other than $K_2$ admits an antimagic labelling. In 2017, two sets of authors (Arumugam, Premalatha, Ba\\v{c}a \\& Semani\\v{c}ov\\'{a}-Fe\\v{n}ov\\v{c}\\'{i}kov\\'{a} and Bensmail, Senhaji \\& Szabo Lyngsie) independently introduced the weaker notion of a local antimagic labelling, where only adjacent vertices must be distinguished. Both sets of authors conjectured that any connected graph other than $K_2$ admits a local antimagic labelling. Recently, Haslegrave (2018) proved that every graph without isolated edges admits a local antimagic labeling. A graph $G = (V, E)$ is said to be local antimagic total if there exists a bijection $f: V\\cup E \\to\\{1,2,\\ldots ,|V\\cup E|\\}$ such that for any pair of adjacent vertices $u$ and $v$, $w(u)\\not= w(v)$, where the induced vertex weight $w(u)= f(u) +\\sum f(e)$, with $e$ ranging over all the edges incident to $u$. The local antimagic total chromatic number of $G$, denoted by $\\chi_{lat}(G)$, is the minimum number of distinct induced vertex weights over all local antimagic total labelings of $G$. In this note, we proved that every graph admits a local antimagic total labeling. We then determine the exact value of $\\chi_{lat}(G)$ for some standard graphs $G$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-06-25T05:57:28.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C78",
      "05C69"
    ],
    "keywords": [
      "local antimagic total labeling",
      "local antimagic total chromatic number",
      "induced vertex weight",
      "local antimagic labelling",
      "adjacent vertices"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 4,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}