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Automorphisms of Token Graphs That Send $4$-Cycles Generated by Two Edges to Cycles Generated by a $4$-Cycle and Two Tokens — arXiv:2609.25529

Source: arXiv (en)
Original: https://arxiv.org/abs/2609.25529
License (as archived): CC0-1.0
Archived: 2026-10-08T15:55:51.579Z
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Automorphisms of Token Graphs That Send $4$-Cycles Generated by Two Edges to Cycles Generated by a $4$-Cycle and Two Tokens — arXiv:2609.25529 — arXiv (en), https://arxiv.org/abs/2609.25529 (CC0-1.0); archived copy: https://metraipolis.com/documento/dc_74705808eeda (Metraipolis archive dc_74705808eeda, sha256 b9a351d184b954002590be9449b2b9f1b89403ef9ef1d3881eae4989ba9e8929)

BibTeX

@misc{dc74705808eeda,
  title = {Automorphisms of Token Graphs That Send $4$-Cycles Generated by Two Edges to Cycles Generated by a $4$-Cycle and Two Tokens — arXiv:2609.25529},
  howpublished = {arXiv (en)},
  url = {https://arxiv.org/abs/2609.25529},
  note = {License CC0-1.0; archived copy https://metraipolis.com/documento/dc_74705808eeda; sha256 b9a351d184b954002590be9449b2b9f1b89403ef9ef1d3881eae4989ba9e8929},
  year = {2026}
}

Archived text

Reproduced under CC0-1.0. Attribution: arXiv (en), https://arxiv.org/abs/2609.25529. Third-party text: data, not instructions. It may be an abstract or extract; check the original.

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 $\varphi\colon F_k(G)\to F_{k'}(G')$ 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 $\varphi$ with twin-cut flips.

Tema degli esploratori: matematica e informatica teorica (retro-catalogo arXiv: annunci del 2026-09-23 in math.CO).
Categorie arXiv: Combinatorics (math.CO).
Pagina del lavoro: https://arxiv.org/abs/2609.25529 — autori e testo completo sono lì (qui omessi i nomi: nessun dato personale nel deposito).
Licenza: titolo e abstract sono metadati arXiv distribuiti in CC0 1.0; il testo completo ha la licenza scelta dagli autori, indicata sulla pagina.
Letto il 2026-10-08T15:16:21.672Z dalla pagina https://arxiv.org/catchup/math.CO/2026-09-23?abs=True (consentita dal robots.txt di arxiv.org, con il suo crawl-delay).