{
  "id": "1208.4995",
  "version": "v1",
  "published": "2012-08-24T15:01:39.000Z",
  "updated": "2012-08-24T15:01:39.000Z",
  "title": "A characterization of the edge connectivity of direct products of graphs",
  "authors": [
    "Simon Spacapan"
  ],
  "categories": [
    "math.CO"
  ],
  "abstract": "The direct product of graphs $G=(V(G),E(G))$ and $H=(V(H),E(H))$ is the graph, denoted as $G\\times H$, with vertex set $V(G\\times H)=V(G)\\times V(H)$, where vertices $(x_1,y_1)$ and $(x_2,y_2)$ are adjacent in $G\\times H$ if $x_1x_2\\in E(G)$ and $y_1y_2\\in E(H)$. The edge connectivity of a graph $G$, denoted as $\\lambda(G)$, is the size of a minimum edge-cut in $G$. We introduce a function $\\psi$ and prove the following formula %for the edge-connectivity of direct products $$\\lambda (G\\times H)=\\min {2\\lambda(G)|E(H)|,2\\lambda(H)|E(G)|,\\delta(G\\times H), \\psi(G,H), \\psi(H,G)} .$$ We also describe the structure of every minimum edge-cut in $G\\times H$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2012-08-24T15:01:39.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "direct product",
      "edge connectivity",
      "minimum edge-cut",
      "characterization"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2012arXiv1208.4995S"
    }
  }
}