On the expansion of Hanoi graphs
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arXiv
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| Auteurs principaux: | , , |
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866915565757530112 |
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| author | Eppstein, David Frishberg, Daniel Maxwell, William |
| author_facet | Eppstein, David Frishberg, Daniel Maxwell, William |
| contents | 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 $H_p^n$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_18010 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the expansion of Hanoi graphs Eppstein, David Frishberg, Daniel Maxwell, William Combinatorics Discrete Mathematics 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 $H_p^n$. |
| title | On the expansion of Hanoi graphs |
| topic | Combinatorics Discrete Mathematics |
| url | https://arxiv.org/abs/2510.18010 |