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

Taizo Sadahiro

TL;DR

The paper addresses the Hamiltonicity of the state graph for the cylindrical $5$-puzzle on a toroidal $2\times 3$ grid, proving the existence of an explicit Hamiltonian cycle on a $720$-vertex graph. It introduces a quotient-lifting framework that reduces the problem to a smaller symmetry quotient, solves the reduced problem, and lifts the solution back to the full Cayley-state graph, producing compact, human-readable cycle encodings. The main results include a $48$-move cycle encoded over $\{L,R,V\}$ (repeated $15$ times) and a $24$-move word yielding a $2$-cycle cover that can be spliced into a Hamiltonian path, together with a full Hamiltonian cycle on the state graph encoded by a long word $\hat{c}$. The method generalizes to other symmetric configuration spaces, and the quotient constructions reveal a potential combinatorial Hamiltonian grammar and a monodromy explanation for the lifted cycles, with promising extensions to larger puzzles.

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.

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

TL;DR

The paper addresses the Hamiltonicity of the state graph for the cylindrical -puzzle on a toroidal grid, proving the existence of an explicit Hamiltonian cycle on a -vertex graph. It introduces a quotient-lifting framework that reduces the problem to a smaller symmetry quotient, solves the reduced problem, and lifts the solution back to the full Cayley-state graph, producing compact, human-readable cycle encodings. The main results include a -move cycle encoded over (repeated times) and a -move word yielding a -cycle cover that can be spliced into a Hamiltonian path, together with a full Hamiltonian cycle on the state graph encoded by a long word . The method generalizes to other symmetric configuration spaces, and the quotient constructions reveal a potential combinatorial Hamiltonian grammar and a monodromy explanation for the lifted cycles, with promising extensions to larger puzzles.

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 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.

Paper Structure

This paper contains 5 sections, 3 theorems, 30 equations, 2 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. State graph as a Cayley graph
  4. Hamiltonian path and cycle
  5. Quotient-Lifting method

Key Result

Lemma 1

Let $s = (s_1, s_2, \ldots, s_{12}) \in \{\mathtt{L}, \mathtt{R}, \mathtt{V}\}^{12}$ be the sequence and let $s' = (s_{13}, s_{14}, \ldots, s_{24})$ be its perturbation defined by Define $w$ as the five-fold repetition of $(s_1,s_2,\ldots, s_{24})$, that is, Then the sequence is a Hamiltonian cycle in the Cayley graph ${\rm Cay}(S_5,\{{\mathtt L}, {\mathtt R}, {\mathtt V}\})$, where $g_i=\phi_

Figures (2)

  • Figure 1: The cylindrical grid $\Gamma_{2,3}$.
  • Figure 2: A Hamiltonian cycle on the quotient graph $K_1\backslash X$ of order $48$ (left side) lifts to a single Hamiltonian cycle of length $720$ in the full state graph ${\rm puz}(\Gamma_{2,3})$ (right side). This visualization illustrates the essence of the quotient-lifting method.

Theorems & Definitions (7)

  • Lemma 1
  • proof
  • Theorem 1
  • proof
  • Theorem 2
  • proof
  • Remark 1