发表机构
CINVESTAV; UNAM（墨西哥国立理工学院计算与系统研究进展中心; 墨西哥国立自治大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究令牌图的自同构，证明若同构将两条边生成的4-环映射到4-环生成的环，则对应图存在孪生割，且扭转可通过孪生割翻转消除。
AI 中文摘要
设 $G$ 为连通图。$G$ 的 $k$-令牌图是图 $F_k(G)$，其顶点集由 $G$ 的 $k$ 个顶点的所有子集组成，当两个子集的对称差是 $G$ 的一条边时，它们相邻。$G$ 的每个自同构都诱导 $F_k(G)$ 的一个自同构，当 $k=|G|/2$ 时，补集映射也是如此；这种形式的自同构称为诱导自同构。Fabila-Monroy 等人（《图与组合》42，2026）表明，令牌图可以有许多非诱导自同构，这些自同构源于孪生割。孪生割是割集 $\{x,y\}$，其两个顶点在 $G$ 中具有相同的邻居（除彼此之外）。这些非诱导自同构将 $G\setminus \{x,y\}$ 的每个分量上具有指定数量令牌（总计 $k-1$）且在 $\{x,y\}$ 的一个顶点上恰好有一个令牌的配置，发送到通过将 $\{x,y\}$ 处的令牌翻转（即移动到 $\{x,y\}$ 的另一个顶点）而得到的配置。这些自同构将由在两条不相交边上移动两个令牌生成的诱导 $4$-环发送到由在一个 $4$-环上移动两个令牌生成的环；由此引入了我们称之为扭转的现象。我们证明了一个部分逆命题：如果同构 $\varphi\colon F_k(G)\to F_{k'}(G')$ 将由在两条不相交边上移动两个令牌生成的某个 $4$-环发送到由在一个 $4$-环上移动两个令牌生成的环，则 $G$ 和 $G'$ 分别具有孪生割 $\{x,y\}$ 和 $\{x',y'\}$。我们还表明，任何扭转都可以通过将 $\varphi$ 与孪生割翻转复合来消除。
英文摘要
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.
