On the expansion of Hanoi graphs

TL;DR

This work analyzes the expansion and treewidth of Hanoi graphs $H_p^n$ for fixed $p\ge 3$, building on prior results that bounded the treewidth upper bound by $O((p-2)^n)$ and provided partial lower bounds. The authors extend a Markov chain decomposition framework to graphs without a natural Markov chain interpretation, adapting it to the Hanoi graph structure by recursively partitioning into $p$ copies of $H_p^{n-1}$ and introducing a refined distribution step that leverages facet-based boundary decompositions. They prove a tight lower bound on expansion, showing $h(H_p^n) = \Theta(((p-2)/p)^n)$, which immediately implies the treewidth bound $tw(H_p^n) = \Theta((p-2)^n)$ via known relations between expansion and treewidth. The distribution mechanism combines a sequence of multicommodity-flow subproblems (shuffling, concentration, transmission, and distribution) within a hierarchical, facet-aware routing scheme, enabling a controlled, recursive construction of flows and overcoming obstacles posed by sparse inter-subgraph connections. Overall, the paper provides tight expansion bounds for Hanoi graphs and demonstrates how a refined decomposition framework can be applied to non-MCMC graph families with significant algorithmic implications.

Abstract

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 $Ω(n^{(1-p)/2}(p-2)^n)$ 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 $Ω((p-2)^n)$ for the expansion of $H_p^n$.

Figures (5)

• Figure 1: The Hanoi graphs $H_3^3$ (left), $H_4^2$ (center), and $H_4^4$ (right), with their vertices arranged recursively in subgraphs. At the top level, the three (or four) subgraphs in each graph are induced by placements of the largest disc in the puzzle. In $H_4^4$, we depict in different colors the boundary matchings at different recursive levels of the graph. At the top two levels, we omit individual edges, and instead highlight the boundary sets in magenta (top level) and turquoise (second level).
• Figure 2: The transmission and distribution steps for $H_3^3$ as described in Section \ref{['sec:hanoiexplb']}. The top vertex in $H_1$ routes $|V(H_1)||V(H_2)|$ flow to its neighbor in $H_2$. That neighbor, the bottom-left vertex in $H_2$, then routes $|V(H_1)|\cdot 3$ flow to its topmost neighbor (similarly right neighbor) that is bound for the topmost (similarly bottom-right) 3-clique in $H_2$, and distributes an additional $|V(H_1)|\cdot 1$ flow to each neighbor in its own 3-clique.
• Figure 3: A schematic illustration showing the decomposition of the Hanoi subgraphs into subgraphs. $H_{ij}$ is the subgraph induced by the set of configurations in which the largest disc is placed on peg $i$ and the second-largest disc is placed on peg $j$.
• Figure 4: Top left: The problem $\pi_{dist}$ is the problem of distributing flow received from $H_1$ throughout $H_2$. Since $\partial_v(H_2,H_1) = \mathcal{F}_{12}(H_2) = \mathcal{F}_{12}(H_{23}) \cup \mathcal{F}_{12}(H_{24})$, this flow is initially concentrated in the facets $\mathcal{F}_{12}(H_{23})$ and $\mathcal{F}_{12}(H_{24})$ and must be distributed throughout $H_{2}$. Top right: The problems $\pi_{distsub,3}$ and $\pi_{distsub,4}$ of distributing flow received from $H_1$ throughout $H_{23}$ and $H_{24}$. This flow is initially concentrated within $\mathcal{F}_{12}(H_{23})$ and $\mathcal{F}_{12}(H_{24})$. Bottom: $\pi_{tran,31}$ is a subproblem of $\pi_{distsub,3}$ and involves sending the $H_{23}\rightarrow H_{21}$ flow (which originates at $H_1$) across the boundary matching $\partial_e(H_{23},H_{21})$.
• Figure 5: An illustration of the "routing" problems $\pi_{rout,i,1}, \pi_{rout,i,2}$ in \ref{['lem:fdistrecdef']}.

Theorems & Definitions (32)

• Theorem 1.1
• Corollary 1.2
• Definition 2.1
• Remark 2.2
• Lemma 2.3
• proof
• Lemma 2.3
• Lemma 2.4
• Corollary 2.5
• Remark 2.6