{
  "id": "1407.8035",
  "version": "v2",
  "published": "2014-07-30T13:27:46.000Z",
  "updated": "2015-11-30T14:25:41.000Z",
  "title": "On Chromatic Number and Minimum Cut",
  "authors": [
    "Meysam Alishahi",
    "Hossein Hajiabolhassan"
  ],
  "categories": [
    "math.CO"
  ],
  "abstract": "For a graph $G$, the tree graph ${\\cal T}_{G,t}$ has all tree subgraphs of $G$ with $t$ vertices as vertex set and two tree subgraphs are neighbors if they are edge-disjoint. Also, the $r^{th}$ cut number of $G$ is the minimum number of edges between parts of a partition of vertex set of $G$ into two parts such that each part has size at least $r$. We show that if $t=(1-o(1))n$ and $n$ is large enough, then for any dense graph $G$ with $n$ vertices, the chromatic number of the tree graph ${\\cal T}_{G,t}$ is equal to the $(n-t+1)^{th}$ cut number of $G$. In particular, as a consequence, we prove that if $n$ is large enough and $G$ is a dense graph, then the chromatic number of the spanning tree graph ${\\cal T}_{G,n}$ is equal to the size of the minimum cut of $G$. The proof method is based on alternating Tur\\'an number inspired by Tucker's lemma, an equivalent combinatorial version of the Borsuk-Ulam theorem.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-07-30T13:27:46.000Z",
      "abstract": "For a graph $G$, the tree graph ${\\cal T}_{G,t}$ has all tree subgraphs of $G$ with $t$ vertices as vertex set and two tree subgraphs are neighbors if they are edge-disjoint. Also, the $r^{th}$ cut number of $G$ is the minimum number of edges between parts of a partition of vertex set of $G$ into two parts such that each part has size at least $r$. We show that if $t=(1-o(1))n$ and $n$ is large enough, then for any dense graph $G$ with $n$ vertices, the chromatic number of the tree graph ${\\cal T}_{G,t}$ is equal to the $(n-t+1)^{th}$ cut number of $G$. In particular, as a consequence, we prove that if $n$ is large enough and $G$ is a dense graph, then the chromatic number of the spanning tree graph ${\\cal T}_{G,n}$ is equal to the size of the minimum cut of $G$. The proof method is based on alternating chromatic number inspired by Tucker's lemma, an equivalent combinatorial version of the Borsuk-Ulam theorem.",
      "comment": null,
      "journal": null,
      "doi": null
    },
    {
      "version": "v2",
      "updated": "2015-11-30T14:25:41.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C15"
    ],
    "keywords": [
      "chromatic number",
      "minimum cut",
      "tree graph",
      "tree subgraphs",
      "dense graph"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1407.8035A"
    }
  }
}