Hamilton decompositions of all directed tori at odd modulus
Fuente:
arXiv
Guardado en:
| Autor principal: | |
|---|---|
| Formato: | Preprint |
| Publicado: |
2026
|
| Materias: | |
| Acceso en línea: | |
| Etiquetas: |
Agregar Etiqueta
Sin Etiquetas, Sea el primero en etiquetar este registro!
|
| _version_ | 1866913106046746624 |
|---|---|
| author | Park, SangHyun |
| author_facet | Park, SangHyun |
| contents | Let $D_d(m) = \mathrm{Cay}((\mathbb{Z}/m\mathbb{Z})^d, {e_0, \ldots, e_{d-1}})$ denote the directed Cayley graph on the positive coordinate basis, equivalently the Cartesian product of $d$ directed cycles of length $m$. The equal side directed Hamilton decomposition problem asks when the arc set of $D_d(m)$ partitions into $d$ directed Hamilton cycles. We prove that such a decomposition exists for every $d \geq 2$ and every odd $m \geq 3$, settling the equal side directed Hamilton decomposition problem at all odd moduli. The proof combines root flat certificate theorem, a prefix count primitivity criterion, and a modular trade lifting theorem with two closure principles: the Cartesian product and the successor step $b \mapsto 2b+1$. Together these propagate the small base dimensions $d \in {2, 3, 5, 7}$ to all $d \geq 2$. The boundary cases $D_7(3)$ and $D_7(5)$, where the prefix-count family saturates its zero symbol budget, are handled by explicit non prefix zero set root flat certificates whose zero set compiler. An accompanying Lean 4 formalization verifies the main theorem and the finite certificate predicates. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_04734 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Hamilton decompositions of all directed tori at odd modulus Park, SangHyun Combinatorics Discrete Mathematics Logic in Computer Science 05C45, 05C70 Let $D_d(m) = \mathrm{Cay}((\mathbb{Z}/m\mathbb{Z})^d, {e_0, \ldots, e_{d-1}})$ denote the directed Cayley graph on the positive coordinate basis, equivalently the Cartesian product of $d$ directed cycles of length $m$. The equal side directed Hamilton decomposition problem asks when the arc set of $D_d(m)$ partitions into $d$ directed Hamilton cycles. We prove that such a decomposition exists for every $d \geq 2$ and every odd $m \geq 3$, settling the equal side directed Hamilton decomposition problem at all odd moduli. The proof combines root flat certificate theorem, a prefix count primitivity criterion, and a modular trade lifting theorem with two closure principles: the Cartesian product and the successor step $b \mapsto 2b+1$. Together these propagate the small base dimensions $d \in {2, 3, 5, 7}$ to all $d \geq 2$. The boundary cases $D_7(3)$ and $D_7(5)$, where the prefix-count family saturates its zero symbol budget, are handled by explicit non prefix zero set root flat certificates whose zero set compiler. An accompanying Lean 4 formalization verifies the main theorem and the finite certificate predicates. |
| title | Hamilton decompositions of all directed tori at odd modulus |
| topic | Combinatorics Discrete Mathematics Logic in Computer Science 05C45, 05C70 |
| url | https://arxiv.org/abs/2605.04734 |