Abstract: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 $m\ge3$ and $d\ge2$. 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 $d\ge7$, 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.
From: SangHyun Park [view email]
[v1]
Wed, 6 May 2026 10:32:39 UTC (32 KB)
[v2]
Sun, 10 May 2026 06:51:16 UTC (58 KB)
[v3]
Thu, 17 Sep 2026 05:54:51 UTC (60 KB)
此内容由惯性聚合(RSS阅读器)自动聚合整理，仅供阅读参考。 原文来自 — 版权归原作者所有。