{
  "id": "2401.12347",
  "version": "v1",
  "published": "2024-01-22T20:26:12.000Z",
  "updated": "2024-01-22T20:26:12.000Z",
  "title": "On a problem concerning integer distance graphs",
  "authors": [
    "Janka Oravcová",
    "Roman Soták"
  ],
  "categories": [
    "math.CO"
  ],
  "abstract": "For $D$ being a subset of positive integers, the integer distance graph is the graph $G(D)$, whose vertex set is the set of integers, and edge set is the set of all pairs $uv$ with $|u-v| \\in D$. It is known that $\\chi(G(D)) \\leq |D|+1$. This article studies the problem (which is motivated by a conjecture of Zhu): \"Is it true that $\\chi(G(D)) = |D|+1$ implies $\\omega(G(D)) \\geq |D|+1$, where $\\omega(H)$ is the clique number of $H$?\". We give a negative answer to this question, by showing an infinite class of integer distance graphs with $\\chi(G(D))=|D|+1$ but $\\omega(G(D))=|D|-1$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-01-22T20:26:12.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C15",
      "05C63",
      "05C75"
    ],
    "keywords": [
      "problem concerning integer distance graphs",
      "vertex set",
      "edge set",
      "article studies",
      "clique number"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}