{
  "id": "1508.07485",
  "version": "v1",
  "published": "2015-08-29T17:47:35.000Z",
  "updated": "2015-08-29T17:47:35.000Z",
  "title": "On zero-sum $\\mathbb{Z}_{2j}^k$-magic graphs",
  "authors": [
    "J. P. Georges",
    "D. Mauro",
    "K. Wash"
  ],
  "categories": [
    "math.CO"
  ],
  "abstract": "Let $G = (V,E)$ be a finite graph and let $(\\mathbb{A},+)$ be an abelian group with identity 0. Then $G$ is \\textit{$\\mathbb{A}$-magic} if and only if there exists a function $\\phi$ from $E$ into $\\mathbb{A} - \\{0\\}$ such that for some $c \\in \\mathbb{A}$, $\\sum_{e \\in E(v)} \\phi(e) = c$ for every $v \\in V$, where $E(v)$ is the set of edges incident to $v$. Additionally, $G$ is \\textit{zero-sum $\\mathbb{A}$-magic} if and only if $\\phi$ exists such that $c = 0$. We consider zero-sum $\\mathbb{A}$-magic labelings of graphs, with particular attention given to $\\mathbb{A} = \\mathbb{Z}_{2j}^k$. For $j \\geq 1$, let $\\zeta_{2j}(G)$ be the smallest positive integer $c$ such that $G$ is zero-sum $\\mathbb{Z}_{2j}^c$-magic if $c$ exists; infinity otherwise. We establish upper bounds on $\\zeta_{2j}(G)$ when $\\zeta_{2j}(G)$ is finite, and show that $\\zeta_{2j}(G)$ is finite for all $r$-regular $G$, $r \\geq 2$. Appealing to classical results on the factors of cubic graphs, we prove that $\\zeta_4(G) \\leq 2$ for a cubic graph $G$, with equality if and only if $G$ has no 1-factor. We discuss the problem of classifying cubic graphs according to the collection of finite abelian groups for which they are zero-sum group-magic.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-08-29T17:47:35.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C78"
    ],
    "keywords": [
      "magic graphs",
      "finite abelian groups",
      "magic labelings",
      "zero-sum group-magic",
      "establish upper bounds"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}