The Edge-connectivity of Token Graphs

Abstract

Let $G$ be a simple graph of order $n\geq 2$ and let $k\in \{1,\ldots ,n-1\}$. The $k$-token graph $F_k(G)$ of $G$ is the graph whose vertices are the $k$-subsets of $V(G)$, where two vertices are adjacent in $F_k(G)$ whenever their symmetric difference is an edge of $G$. In 2018 J. Lea\~nos and A. L. Trujillo-Negrete proved that if $G$ is $t$-connected and $t\geq k$, then $F_k(G)$ is at least $k(t-k+1)$-connected. In this paper we show that such a lower bound remains true in the context of edge-connectivity. Specifically, we show that if $G$ is $t$-edge-connected and $t\geq k$, then $F_k(G)$ is at least $k(t-k+1)$-edge-connected. We also provide some families of graphs attaining this bound.