On the Treewidth of Token and Johnson Graphs

Let $G$ be a graph on $n$ vertices and $1 \le k \le n$ a fixed integer. The \textit{$k$-token graph} of $G$ is the graph $F_k(G)$ whose vertex set consists of all $k$-subsets of the vertex set of $G$, where two vertices $A$ and $B$ are adjacent in $F_k(G)$ whenever their symmetric difference $A\triangle B$ is an edge of $G$. In this paper we study the treewidth of $F_k(G)$ when $G$ is a star, path, or a complete graph. We show that in the first two cases, the treewidth is of order $\Theta(n^{k-1})$, and of order $\Theta(n^k)$ in the third case. We conjecture that our upper bound for the treewidth of $F_k(K_n)$ is tight. This is particularly relevant since $F_k(K_n)$ is isomorphic to the well known Johnson graph $J(n,k)$.