Hamilton decompositions of all directed tori at odd modulus

TL;DR

This paper proves that directed Cayley graphs on odd moduli can be decomposed into directed Hamilton cycles for all dimensions, solving a longstanding problem in graph theory.

Contribution

It establishes the existence of Hamilton decompositions for all directed tori at odd moduli, extending previous results to all dimensions.

Findings

Decomposition exists for all d ≥ 2 and odd m ≥ 3.

The proof combines algebraic certificates and closure principles.

Explicit certificates verify boundary cases D_7(3) and D_7(5).

Abstract

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.