Hamilton decompositions of equal-side directed tori

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 $m\ge 3$ and $d\ge 2$. 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 $d\ge 7$, 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.

44 pages. Substantially revised and extended to all moduli $m\ge 3$ and all dimensions $d\ge 2$. Includes a shorter integer-selection proof for odd moduli, the even-modulus construction, and an ancillary Python script for explicit finite checks