On the expansion of Hanoi graphs

2025-10-20

AI Summary

This paper investigates the graph expansion of the Hanoi graph $H_p^n$ ($p geq 3$), aiming to determine a tight lower bound to verify whether the known upper bound $O((p-2)^n)$ is asymptotically tight. Leveraging the recursive self-similarity of $H_p^n$, the authors combine graph-theoretic expansion analysis, inductive decomposition, and combinatorial counting to rigorously establish an expansion lower bound of $Omega((p-2)^n)$. Consequently, the exact asymptotic order of expansion is settled as $Theta((p-2)^n)$, resolving this long-standing open problem. Furthermore, the work uncovers a fundamental relationship between expansion and treewidth, thereby enriching the theoretical characterization of structural properties of Hanoi graphs. These results provide a critical foundation for analyzing connectivity of state-transition graphs and for modeling parallel computation architectures.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Search and Optimization: Combinatorial Optimization
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Web Mining and Content Analysis: Models for Web evolution
• Security and Privacy: Large-scale security measurements

Abstract

The famous Tower of Hanoi puzzle involves moving $n$ discs of distinct sizes from one of $pgeq 3$ 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 $H_p^n$.

Problem

• Proving tight lower bound for Hanoi graph expansion
• Analyzing asymptotic treewidth bounds for multi-peg puzzles
• Establishing matching Ω((p-2)^n) bound for H_p^n

Innovation

• Proved matching lower bound for expansion
• Used asymptotic analysis of Hanoi graphs
• Applied treewidth bounds to graph expansion