{
  "id": "2012.03259",
  "version": "v1",
  "published": "2020-12-06T13:21:54.000Z",
  "updated": "2020-12-06T13:21:54.000Z",
  "title": "Connectivity of orientations of 3-edge-connected graphs",
  "authors": [
    "Florian Hörsch",
    "Zoltán Szigeti"
  ],
  "comment": "to appear in European Journal of Combinatorics",
  "categories": [
    "math.CO"
  ],
  "abstract": "We attempt to generalize a theorem of Nash-Williams stating that a graph has a $k$-arc-connected orientation if and only if it is $2k$-edge-connected. In a strongly connected digraph we call an arc {\\it deletable} if its deletion leaves a strongly connected digraph. Given a $3$-edge-connected graph $G$, we define its Frank number $f(G)$ to be the minimum number $k$ such that there exist $k$ orientations of $G$ with the property that every edge becomes a deletable arc in at least one of these orientations. We are interested in finding a good upper bound for the Frank number. We prove that $f(G)\\leq 7$ for every $3$-edge-connected graph. On the other hand, we show that a Frank number of $3$ is attained by the Petersen graph. Further, we prove better upper bounds for more restricted classes of graphs and establish a connection to the Berge-Fulkerson conjecture. We also show that deciding whether all edges of a given subset can become deletable in one orientation is NP-complete.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-12-06T13:21:54.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "orientation",
      "frank number",
      "connectivity",
      "strongly connected digraph",
      "edge-connected graph"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}