论Token图与Johnson图的树宽
On the Treewidth of Token and Johnson Graphs
AI总结：
本文研究星图、路径图和完全图的$k$-token图的树宽，得出对应阶数结果，猜想完全图$k$-token图树宽上界紧，且其与Johnson图同构。
摘要：
设$G$是具有$n$个顶点的图，$1 \u2264 k \u2264 n$是固定整数。$G$的\textit{$k$-token图}为图$F_k(G)$，其顶点集由$G$顶点集的所有$k$-子集构成，当且仅当两个顶点$A$和$B$的对称差$A\triangle B$是$G$的一条边时，$A$和$B$在$F_k(G)$中相邻。本文研究当$G$为星图、路径图或完全图时$F_k(G)$的树宽。我们证明在前两种情况下，树宽的阶为$Θ(n^{k-1})$，第三种情况下为$Θ(n^k)$。我们猜想对$F_k(K_n)$树宽的上界是紧的，这尤为重要，因为$F_k(K_n)$与著名的Johnson图同构。
英文摘要：
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 $J(n,k)$.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
