{
  "abstract": "Given graphs $X,Y$ on $n$ vertices, Defant and Kravitz defined the friends and strangers graph, denoted $\\mathrm{FS}(X,Y)$. The vertices of $\\mathrm{FS}(X,Y)$ are bijections $σ:V(X)\\to V(Y)$, and two bijections $σ$ and $σ'$ are adjacent in $\\mathrm{FS}(X,Y)$ if they differ by a transposition $(ab)$ such that $a$ and $b$ are adjacent in $X$ and $σ(a)$ and $σ(b)$ are adjacent in $Y$. Our main result resolves a conjecture of Bangachev that if $X,Y$ are connected graphs and $2\\min\\{δ(X),δ(Y)\\}+3\\max\\{δ(X),δ(Y)\\}\\ge 3n$, then $\\mathrm{FS}(X,Y)$ is connected. The main step in the proof is showing that there exists an absolute constant $C$ such that for any adjacent vertices $a,b\\in V(X)$ and any vertex $σ\\in V(\\mathrm{FS}(X,Y))$, $a$ and $b$ can swap positions in at most $C$ moves. Furthermore, we prove that if $X,Y$ are connected graphs satisfying the above minimum degree condition, then the diameter of $\\mathrm{FS}(X,Y)$ is $O(n^2/\\min\\{δ(X),δ(Y)\\})$, and that this is sharp as a bound on the maximum diameter of $\\mathrm{FS}(X,Y)$ in this regime. This work improves the diameter results of Akella and Li, and resolves many of their questions. Moreover, we show that even for disconnected graphs $X$ and $Y$, the same minimum degree condition implies that the diameter of any connected component of $\\mathrm{FS}(X,Y)$ is $O(n^2)$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.20648",
  "author_total": 1,
  "authors": [
    {
      "name": "Kai Mawhinney"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/fgct4frqxp04g.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.20648",
  "fulltext_url": null,
  "id": "fgct4frqxp04g",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.20648",
  "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"
    ],
    [
      "openalex",
      "1703|Advanced Graph Theory Research"
    ]
  ],
  "openalex": "W7222423744",
  "pdf_url": "https://arxiv.org/pdf/2610.20648",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-12",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/fgct4frqxp04g.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:54Z",
      "native_id": "2610.20648",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.20648"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Connectivity and Diameter of Friends and Strangers Graphs Using Minimum Degree",
  "title_text": "Connectivity and Diameter of Friends and Strangers Graphs Using Minimum Degree",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:54Z",
  "url": "https://research.draconitic.com/paper/fgct4frqxp04g",
  "venue": "arXiv",
  "versions": []
}