{
  "abstract": "Let $G$ be a connected graph. The $k$-token graph of $G$ is the graph $F_k(G)$ whose vertex set consists of all subsets of $k$ vertices of $G$, where two of them are adjacent whenever their symmetric difference is an edge of $G$. Every automorphism of $G$ induces one of $F_k(G)$, as does complementation when $k=|G|/2$; automorphisms of this form are called \\emph{induced}. Fabila-Monroy et al.\\ (Graphs and Combinatorics 42, 2026) show that token graphs can have many non-induced automorphisms, arising from \\emph{twin cuts}. These are cut sets $\\{x,y\\}$ whose two vertices have the same neighbours (other than themselves) in $G$. These non-induced automorphisms send configurations with a prescribed number of tokens on each component of $G\\setminus \\{x,y\\}$, totalling $k-1$, and exactly one token on one vertex of $\\{x,y\\}$, to the configuration obtained by moving (\\emph{flipping}) the token at $\\{x,y\\}$ to the other vertex of $\\{x,y\\}$. These automorphisms send an induced $4$-cycle generated by moving two tokens on two disjoint edges to one generated by moving two tokens on a $4$-cycle; thus introducing what we call a \\emph{twist}. We prove a partial converse: if an isomorphism $φ\\colon F_k(G)\\to F_{k'}(G')$ sends some $4$-cycle generated by moving two tokens on two disjoint edges to one generated by moving two tokens on a $4$-cycle, then $G$ and $G'$ have twin cuts $\\{x,y\\}$ and $\\{x',y'\\}$, respectively. We also show that any twist can be undone by composing $φ$ with twin-cut flips.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.25529",
  "author_total": 3,
  "authors": [
    {
      "name": "Ruy Fabila-Monroy"
    },
    {
      "name": "Sergio Gerardo Gómez-Galicia"
    },
    {
      "name": "Ana Laura Trujillo-Negrete"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/2vz206dmtdx6q.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.25529",
  "fulltext_url": null,
  "id": "2vz206dmtdx6q",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.25529",
  "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": "W7214101640",
  "pdf_url": "https://arxiv.org/pdf/2609.25529",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-22",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/2vz206dmtdx6q.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:23:43Z",
      "native_id": "2609.25529",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.25529"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Automorphisms of Token Graphs That Send $4$-Cycles Generated by Two Edges to Cycles Generated by a $4$-Cycle and Two Tokens",
  "title_text": "Automorphisms of Token Graphs That Send $4$-Cycles Generated by Two Edges to Cycles Generated by a $4$-Cycle and Two Tokens",
  "type": "preprint",
  "updated_at": "2026-10-30T12:23:43Z",
  "url": "https://research.draconitic.com/paper/2vz206dmtdx6q",
  "venue": "arXiv",
  "versions": []
}