{
  "id": "1911.00867",
  "version": "v1",
  "published": "2019-11-03T11:12:19.000Z",
  "updated": "2019-11-03T11:12:19.000Z",
  "title": "On the Standard (2,2)-Conjecture",
  "authors": [
    "Jakub Przybyło"
  ],
  "comment": "13 pages",
  "categories": [
    "math.CO",
    "cs.DM"
  ],
  "abstract": "The well-known 1-2-3 Conjecture asserts that the edges of every graph without an isolated edge can be weighted with $1$, $2$ and $3$ so that adjacent vertices receive distinct weighted degrees. This is open in general. We prove that every graph with minimum degree $\\delta\\geq 10^6$ can be decomposed into two subgraphs requiring just weights $1$ and $2$ for the same goal. We thus prove the so-called Standard $(2,2)$-Conjecture for graphs with sufficiently large minimum degree. The result is in particular based on applications of the Lov\\'asz Local Lemma and theorems on degree-constrained subgraphs.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-11-03T11:12:19.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "adjacent vertices receive distinct",
      "vertices receive distinct weighted degrees",
      "lovasz local lemma",
      "sufficiently large minimum degree",
      "conjecture asserts"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 13,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}