{
  "id": "2309.02993",
  "version": "v1",
  "published": "2023-09-06T13:36:39.000Z",
  "updated": "2023-09-06T13:36:39.000Z",
  "title": "Counting triangles in regular graphs",
  "authors": [
    "Jialin He",
    "Xinmin Hou",
    "Jie Ma",
    "Tianying Xie"
  ],
  "categories": [
    "math.CO"
  ],
  "abstract": "In this paper, we investigate the minimum number of triangles, denoted by $t(n,k)$, in $n$-vertex $k$-regular graphs, where $n$ is an odd integer and $k$ is an even integer. The well-known Andr\\'asfai-Erd\\H{o}s-S\\'os Theorem has established that $t(n,k)>0$ if $k>\\frac{2n}{5}$. In a striking work, Lo has provided the exact value of $t(n,k)$ for sufficiently large $n$, given that $\\frac{2n}{5}+\\frac{12\\sqrt{n}}{5}<k<\\frac{n}{2}$. Here, we bridge the gap between the aforementioned results by determining the precise value of $t(n,k)$ in the entire range $\\frac{2n}{5}<k<\\frac{n}{2}$. This confirms a conjecture of Cambie, de Verclos, and Kang.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-09-06T13:36:39.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "regular graphs",
      "counting triangles",
      "odd integer",
      "minimum number",
      "exact value"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}