On the Treewidth of Token and Johnson Graphs

TL;DR

The paper investigates the treewidth of the $k$-token graph $F_k(G)$ for natural families $G$ with $n$ vertices, focusing on stars, paths, and complete graphs (Johnson graphs). It combines bramble theory, algebraic connectivity, and explicit decompositions to derive both lower and upper bounds, obtaining tight asymptotics for stars and paths: $\tw(F_k(S_n))=\tw(F_k(P_n))=\Theta(n^{k-1})$, and exact $\tw(F_2(S_n))=n-1$ and $\tw(F_2(P_n))=\left\lfloor n/2\right\rfloor$. For $F_2(K_n)$ (the Johnson graph), the exact $\tw$ and $\pw$ are determined by parity, $\tw(F_2(K_n))=\pw(F_2(K_n))=\begin{cases}\frac{n}{2}(\frac{n}{2}-1)+n-2,& n\text{ even}\\(\frac{n-1}{2})^2+n-2,& n\text{ odd}\end{cases}$, and a general upper bound for $\tw(F_k(K_n))$ is provided. The results extend to general fixed $k$, establishing $\tw(F_k(K_n))=\Theta(n^k)$ with sharp bounds for small $k$, and motivate conjectures about optimality for trees and Johnson graphs. Together, these findings deepen understanding of token/Johnson graphs and offer techniques for analyzing treewidth via brambles, spectral methods, and explicit decompositions.

Abstract

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 $Θ(n^{k-1})$, and of order $Θ(n^k)$ 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 $J(n,k)$.

Figures (4)

• Figure 1: A graph $G$ and its $2$-token graph $F_2(G)$.
• Figure 2: A graph $G$ (left) and an optimal tree decomposition $(T, \mathcal{V})$ (right). The colors in the decomposition indicate the bags that contain the respective vertices in $G$, ensuring that $\operatorname{tw}(G) = 3$.
• Figure 3: The $2$-token graph $F_2(S_4)$ of the star $S_4$ (left) and an optimal tree decomposition $(T, \mathcal{V})$ (right). The colors in the decomposition indicate the bags that contain the respective vertices in $F_2(S_4)$.
• Figure 4: Subsets $\mathcal{D}_i$ in $F_2(P_6)$.

Theorems & Definitions (39)

• Theorem 1
• Theorem 2
• Theorem 3
• Theorem 4
• Corollary 4
• Theorem 5: Seymour_Thomas
• Theorem 6: Chandran
• Theorem 6
• proof
• Proposition 7