Hamilton decomposition of odd directed five-tori

For odd m≥3, can the directed Cayley graph of (Z/mZ)^5 with its five positive coordinate generators be decomposed into five directed Hamilton cycles, each visiting every vertex exactly once?

Why it matters and what counts as progress

Why it matters: A five-dimensional infinite family extends structured Hamilton decompositions of directed product graphs.

What counts as progress: Construct a decomposition for every odd m or extend the return-map construction to new torus dimensions.

How the problem developed

29 Apr 2026 · Exact result [1]
SangHyun Park Proved that the directed five-dimensional torus D₅(m) has a Hamilton decomposition for every odd m≥3, using a finite exact-cover certificate and a Lean formalization.
17 Sept 2026 · Generalization [2]
SangHyun Park Extended the construction to all equal-side directed tori D_d(m), for every dimension d≥2 and modulus m≥3, including even moduli.