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

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 $\varphi\colon F_k(G)\to F_{k'}(G')$ 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.