Untangling Eisenstein–Jacobi Networks: The Minimal-Switch Method That Uses Only d-1 Switches

Eisenstein-Jacobi (EJ) networks are degree-six quotient-lattice interconnection networks used in distributed and parallel computing. For a generator α = a + bρ, when gcd(a,b)=d>1, the three unit directions split into d cycles, making the problem of finding three edge-disjoint Hamiltonian cycles (EDHC) a cycle-splicing challenge. Prior work relied on rectangular representations and exchange schedules to prove existence. In a new preprint on arXiv (cs.DC/2606.19832), Bader Albader introduces a different approach: a local switch calculus that operates directly in the natural Cayley geometry of the network.

The key technical contribution is a minimal-switch construction: the first two Hamiltonian cycles are built using exactly d-1 intercomponent switches each—fewer than previously known—and the third cycle is obtained as the unused edge complement. The proof separates four ingredients: component-label collapse, anchor cancellation, noncollision of lifted switch representatives, and connected complement incidence. Crucially, the theorem is not proven by finite computational enumeration but by algebraic symbolic certificates, ensuring rigorousness for specific Euclidean-residue parameter ranges. The author provides tables and CSV data only for verification and reproducibility, not as proof. This work reduces switch complexity in EJ network routing, potentially enabling more efficient message distribution or fault-tolerant designs in high-performance computing topologies.