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arXiv · 2507.03926

A quotient-lifting approach to the Hamiltonicity of the cylindrical 5-puzzle graph

Taizo Sadahiro
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Abstract

We construct an explicit Hamiltonian cycle in the state graph of the 5-puzzle on a toroidal 2x 3 grid, a graph with 720 vertices. The cycle is described by a short symbolic sequence of 48 moves over the alphabet {L,R,V}, repeated $15$ times, which can be verified directly. We also find a shorter 24-move sequence whose repetition yields a 2-cycle cover, which can be spliced into a Hamiltonian path. These constructions arise naturally from a general method: lifting Hamiltonian cycles from a quotient graph under the action of the puzzle's symmetry group. The method produces compact, human-readable cycle encodings and appears effective in broader settings, suggesting a combinatorial grammar underlying Hamiltonian paths in symmetric configuration spaces.

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Taizo Sadahiro. 2025-07-05. A quotient-lifting approach to the Hamiltonicity of the cylindrical 5-puzzle graph. https://arxiv.org/abs/2507.03926

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