The Edge-connectivity of Token Graphs

TL;DR

This paper extends known vertex connectivity results of token graphs to edge connectivity, proving that the edge connectivity of the token graph is at least k(t-k+1) when the original graph is t-edge-connected.

Contribution

It establishes a new lower bound for the edge connectivity of token graphs based on the edge connectivity of the original graph, generalizing previous vertex connectivity results.

Findings

The edge connectivity of the token graph is at least k(t-k+1) when the original graph is t-edge-connected.

The bound is tight for certain families of graphs.

The result generalizes prior vertex connectivity bounds to edge connectivity.

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.