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| Format: | Preprint |
| Published: |
2026
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| Online Access: | https://arxiv.org/abs/2603.24708 |
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| _version_ | 1866914422620946432 |
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| author | Park, SangHyun |
| author_facet | Park, SangHyun |
| contents | We prove that the directed 3-torus D_3(m), or equivalently the Cartesian product of three directed m-cycles, admits a decomposition into three arc-disjoint directed Hamilton cycles for every integer m >= 3. The proof reduces Hamiltonicity to the m-step return maps on the layer section S=i+j+k=0. For odd m, five Kempe swaps of the canonical coloring produce return maps that are explicitly affine-conjugate to the standard 2-dimensional odometer. For even m, a sign-product invariant rules out Kempe-from-canonical constructions, and a different low-layer witness reduces after one further first-return map to a finite-defect clock-and-carry system. The remaining closure is a finite splice analysis, and the case m=4 is handled separately by a finite witness. A Lean 4 formalization accompanies the construction. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_24708 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Hamilton decompositions of the directed 3-torus: a return-map and odometer view Park, SangHyun Combinatorics Discrete Mathematics Group Theory We prove that the directed 3-torus D_3(m), or equivalently the Cartesian product of three directed m-cycles, admits a decomposition into three arc-disjoint directed Hamilton cycles for every integer m >= 3. The proof reduces Hamiltonicity to the m-step return maps on the layer section S=i+j+k=0. For odd m, five Kempe swaps of the canonical coloring produce return maps that are explicitly affine-conjugate to the standard 2-dimensional odometer. For even m, a sign-product invariant rules out Kempe-from-canonical constructions, and a different low-layer witness reduces after one further first-return map to a finite-defect clock-and-carry system. The remaining closure is a finite splice analysis, and the case m=4 is handled separately by a finite witness. A Lean 4 formalization accompanies the construction. |
| title | Hamilton decompositions of the directed 3-torus: a return-map and odometer view |
| topic | Combinatorics Discrete Mathematics Group Theory |
| url | https://arxiv.org/abs/2603.24708 |