The famous Tower of Hanoi puzzle involves moving $n$ discs of distinct sizes from one of $p\geq 3$ pegs (traditionally $p=3$) to another of the pegs, subject to the constraints that only one disc may be moved at a time, and no disc can ever be placed on a disc smaller than itself. Much is known about the Hanoi graph $H_p^n$, whose $p^n$ vertices represent the configurations of the puzzle, and whose edges represent the pairs of configurations separated by a single legal move. In a previous paper, the present authors presented nearly tight asymptotic bounds of $O((p-2)^n)$ and $\Omega(n^{(1-p)/2}(p-2)^n)$ on the treewidth of this graph for fixed $p \geq 3$. In this paper we show that the upper bound is tight, by giving a matching lower bound of $\Omega((p-2)^n)$ for the expansion of $H_p^n$.

翻译：著名的河内塔谜题涉及将 $n$ 个大小各异的圆盘从 $p\geq 3$ 根柱子（传统上 $p=3$）中的一根移动到另一根，约束条件是每次只能移动一个圆盘，且任何圆盘都不能放置在比它小的圆盘之上。关于河内图 $H_p^n$ 已有大量研究，该图的 $p^n$ 个顶点代表谜题的配置状态，其边则代表通过一次合法移动即可相互转换的配置对。在先前的一篇论文中，本文作者针对固定 $p \geq 3$ 的情形，给出了该图树宽的近乎紧的渐近界 $O((p-2)^n)$ 与 $\Omega(n^{(1-p)/2}(p-2)^n)$。本文通过证明 $H_p^n$ 的扩展具有匹配的下界 $\Omega((p-2)^n)$，表明上述上界是紧的。