{
  "id": "2305.11763",
  "version": "v1",
  "published": "2023-05-19T15:49:54.000Z",
  "updated": "2023-05-19T15:49:54.000Z",
  "title": "Cliques in Squares of Graphs with Maximum Average Degree less than 4",
  "authors": [
    "Daniel W. Cranston",
    "Gexin Yu"
  ],
  "comment": "15 pages, 7 figures",
  "categories": [
    "math.CO"
  ],
  "abstract": "Hocquard, Kim, and Pierron constructed, for every even integer $D\\ge 2$, a 2-degenerate graph $G_D$ with maximum degree $D$ such that $\\omega(G_D^2)=\\frac52D$. They asked whether (a) there exists $D_0$ such that every 2-degenerate graph $G$ with maximum degree $D\\ge D_0$ satisfies $\\chi(G^2)\\le \\frac52D$ and (b) whether this result holds more generally for every graph $G$ with mad(G)<4. In this direction, we prove upper bounds on the clique number $\\omega(G^2)$ of $G^2$ that match the lower bound given by this construction, up to small additive constants. We show that if $G$ is 2-degenerate with maximum degree $D$, then $\\omega(G^2)\\le \\frac52D+72$ (with $\\omega(G^2)\\le \\frac52D+60$ when $D$ is sufficiently large). And if $G$ has mad(G)<4 and maximum degree $D$, then $\\omega(G^2)\\le \\frac52D+532$. Thus, the construction of Hocquard et al. is essentially best possible.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-05-19T15:49:54.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C69",
      "05C35",
      "05C15"
    ],
    "keywords": [
      "maximum average degree",
      "maximum degree",
      "upper bounds",
      "small additive constants",
      "clique number"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 15,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}