{
  "id": "2009.06867",
  "version": "v1",
  "published": "2020-09-15T04:55:53.000Z",
  "updated": "2020-09-15T04:55:53.000Z",
  "title": "Group Connectivity under $3$-Edge-Connectivity",
  "authors": [
    "Miaomiao Han",
    "Jiaao Li",
    "Xueliang Li",
    "Meiling Wang"
  ],
  "comment": "to appear in Journal of Graph Theory",
  "categories": [
    "math.CO"
  ],
  "abstract": "Let $S,T$ be two distinct finite Abelian groups with $|S|=|T|$. A fundamental theorem of Tutte shows that a graph admits a nowhere-zero $S$-flow if and only if it admits a nowhere-zero $T$-flow. Jaeger, Linial, Payan and Tarsi in 1992 introduced group connectivity as an extension of flow theory, and they asked whether such a relation holds for group connectivity analogy. It was negatively answered by Hu\\v{s}ek, Moheln\\'{i}kov\\'{a} and \\v{S}\\'{a}mal in 2017 for graphs with edge-connectivity 2 for the groups $S=\\mathbb{Z}_4$ and $T=\\mathbb{Z}_2^2$. In this paper, we extend their results to $3$-edge-connected graphs (including both cubic and general graphs), which answers open problems proposed by Hu\\v{s}ek, Moheln\\'{i}kov\\'{a} and \\v{S}\\'{a}mal(2017) and Lai, Li, Shao and Zhan(2011). Combining some previous results, this characterizes all the equivalence of group connectivity under $3$-edge-connectivity, showing that every $3$-edge-connected $S$-connected graph is $T$-connected if and only if $\\{S,T\\}\\neq \\{\\mathbb{Z}_4,\\mathbb{Z}_2^2\\}$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-09-15T04:55:53.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C21",
      "05C40",
      "05C15"
    ],
    "keywords": [
      "edge-connectivity",
      "distinct finite abelian groups",
      "group connectivity analogy",
      "answers open problems",
      "general graphs"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}