{
  "id": "1906.02854",
  "version": "v1",
  "published": "2019-06-07T01:18:10.000Z",
  "updated": "2019-06-07T01:18:10.000Z",
  "title": "Long monochromatic paths and cycles in $2$-edge-colored graphs with large minimum degree",
  "authors": [
    "József Balogh",
    "Alexandr Kostochka",
    "Mikhail Lavrov",
    "Xujun Liu"
  ],
  "comment": "11 pages",
  "categories": [
    "math.CO"
  ],
  "abstract": "Our main result is a proof for sufficiently large $n$ of the conjecture by Benevides, \\L uczak, Scott, Skokan and White that for every positive integer $n$ of the form $n=3t+r$ where $r \\in \\{0,1,2\\}$ and every $n$-vertex graph $G$ with $\\delta(G) \\ge 3n/4$, in each $2$-edge-coloring of $G$ there exists a monochromatic cycle of length at least $2t+r$. Our result implies the conjecture of Schelp that for every sufficiently large $n$, every $(3n-1)$-vertex graph $G$ with minimum degree larger than $3|V(G)|/4$, in every $2$-edge-coloring of $G$, there is a monochromatic path with $2n$ vertices.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-06-07T01:18:10.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C15",
      "05C35",
      "05C38"
    ],
    "keywords": [
      "large minimum degree",
      "long monochromatic paths",
      "edge-colored graphs",
      "vertex graph",
      "sufficiently large"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 11,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}