{
  "abstract": "For a prime $p$ and $k\\in\\mathbb{F}_p$, the generalized Markoff graph $G_{p,k}$ is an undirected graph whose vertices are the solutions over the finite field $\\mathbb{F}_p$ of the normalized Markoff equation \\[ x_1^2+x_2^2+x_3^2=x_1x_2x_3+k, \\] where two vertices are adjacent if they differ by a Vieta involution. The Markoff graph $G_p$ is obtained from $G_{p,0}$ by removing the origin. The structure of $G_p$ has been the subject of extensive study; in particular, a major breakthrough of Bourgain, Gamburd, and Sarnak established that $G_p$ contains a giant connected component. Combined with Chen's remarkable divisibility theorem, this implies that $G_p$ is connected for all sufficiently large primes $p$. In the same paper, Bourgain, Gamburd, and Sarnak further asked whether the family $\\{G_p\\colon \\text{primes }p\\geq 5\\}$ forms an expander family. This motivates us to investigate the robustness of connectivity in the Markoff graphs. The main result of this paper is proved in the general setting: for every prime $p\\geq5$ and every $k\\in\\mathbb{F}_p\\setminus\\{4\\}$, each connected component $C$ of $G_{p,k}$ with $|V(C)|\\geq 3$ is $2$-connected. Reducing to the case $k=0$, we conclude that if the Markoff graph $G_p$ is connected, then it is in fact $2$-connected. Consequently, the Markoff graph $G_p$ is $2$-connected for all sufficiently large primes $p$. This is sharp in the sense that $G_p$ is not $3$-connected for any prime $p\\geq 7$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.07880",
  "author_total": 3,
  "authors": [
    {
      "name": "Jie Ma"
    },
    {
      "name": "Mengxi Yang"
    },
    {
      "name": "Zichen Yang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/1s65t4hm0tva9.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.07880",
  "fulltext_url": null,
  "id": "1s65t4hm0tva9",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2608.07880",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.CO"
    ],
    [
      "arxiv",
      "math.NT"
    ],
    [
      "openalex",
      "2607|Limits and Structures in Graph Theory"
    ]
  ],
  "openalex": "W7202169718",
  "pdf_url": "https://arxiv.org/pdf/2608.07880",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-03",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/1s65t4hm0tva9.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:18:36Z",
      "native_id": "2608.07880",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.07880"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Sharp vertex connectivity of the Markoff graphs modulo $p$",
  "title_text": "Sharp vertex connectivity of the Markoff graphs modulo p",
  "type": "preprint",
  "updated_at": "2026-10-30T12:18:36Z",
  "url": "https://research.draconitic.com/paper/1s65t4hm0tva9",
  "venue": "arXiv",
  "versions": []
}