{
  "id": "2111.13342",
  "version": "v3",
  "published": "2021-11-26T07:45:25.000Z",
  "updated": "2022-08-27T22:16:03.000Z",
  "title": "On connected components with many edges",
  "authors": [
    "Sammy Luo"
  ],
  "comment": "12 pages, 2 figures; replaced with version revised according to reviewers' feedback. Includes a significantly improved lower bound for the case of a general number of colors",
  "categories": [
    "math.CO"
  ],
  "abstract": "We prove that if $H$ is a subgraph of a complete multipartite graph $G$, then $H$ contains a connected component $H'$ satisfying $|E(H')||E(G)|\\geq |E(H)|^2$. We use this to prove that every three-coloring of the edges of a complete graph contains a monochromatic connected subgraph with at least $1/6$ of the edges. We further show that such a coloring has a monochromatic circuit with a fraction $1/6-o(1)$ of the edges. This verifies a conjecture of Conlon and Tyomkyn. Moreover, for general $k$, we show that every $k$-coloring of the edges of $K_n$ contains a monochromatic connected subgraph with at least $\\frac{1}{k^2-k+\\frac{5}{4}}\\binom{n}{2}$ edges.",
  "revisions": [
    {
      "version": "v3",
      "updated": "2022-08-27T22:16:03.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C15",
      "05C35",
      "05C40",
      "05C55"
    ],
    "keywords": [
      "connected component",
      "monochromatic connected subgraph",
      "complete multipartite graph",
      "complete graph contains",
      "monochromatic circuit"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 12,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}