Hamilton decompositions of equal-side directed tori
Abstract
Let Dd(m) be the Cartesian product of d positively oriented directed cycles of length m. We prove that Dd(m) decomposes into d directed Hamilton cycles for every m≥3 and d≥2. The construction proceeds by splitting coordinate directions in directed multitori. An integer selection theorem supplies unit voltages compatible with the prescribed arc multiplicities; at even modulus, the decisive condition is the parity of each incidence component. For even m and odd d≥7, we satisfy this condition with a factorization having one additional cycle. A relative lifting theorem preserves the first-return data of a three-colour recolouring through successive coordinate splits, after which the recolouring removes the additional cycle on a set whose size is independent of the dimension. Explicit constructions complete the low-dimensional cases. The theorem yields Hamilton decompositions of Cartesian products of equal-order, equal-degree Hamilton-decomposable digraphs, of abelian Cayley digraphs whose generators partition into module bases, and of a family of height-stretched directed tori.
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BibTeX
@misc{park2026hamilton,
title = {{Hamilton decompositions of equal-side directed tori}},
author = {SangHyun Park},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2605.04734},
eprint = {2605.04734},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2605.04734},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Hamilton decompositions of equal-side directed tori
AU - SangHyun Park
PY - 2026
DA - 2026/09/17
PB - arXiv
DO - 10.48550/arxiv.2605.04734
UR - https://arxiv.org/abs/2605.04734
AB - Let D_d(m) be the Cartesian product of d positively oriented directed cycles of length m. We prove that D_d(m) decomposes into d directed Hamilton cycles for every m≥3 and d≥2. The construction proceeds by splitting coordinate directions in directed multitori. An integer selection theorem supplies unit voltages compatible with the prescribed arc multiplicities; at even modulus, the decisive condition is the parity of each incidence component. For even m and odd d≥7, we satisfy this condition with a factorization having one additional cycle. A relative lifting theorem preserves the first-return data of a three-colour recolouring through successive coordinate splits, after which the recolouring removes the additional cycle on a set whose size is independent of the dimension. Explicit constructions complete the low-dimensional cases. The theorem yields Hamilton decompositions of Cartesian products of equal-order, equal-degree Hamilton-decomposable digraphs, of abelian Cayley digraphs whose generators partition into module bases, and of a family of height-stretched directed tori.
ER -