A Unified Constant-Time Switch Rule for Constructing Edge-Disjoint Hamiltonian Cycles in Gaussian Networks

Gaussian networks are degree-four symmetric interconnection networks defined over residue classes of Gaussian integers. Earlier work showed that, when the generator α=a+bi satisfies gcd(a,b)=1, the real and imaginary dimensions directly form two edge-disjoint Hamiltonian cycles. A later construction extended the result to the non-coprime case gcd(a,b)=d>1, but its proof relied on long node-sequence tables and separate odd/even cases for d. This paper presents a unified closed-form construction that covers both d=1 and d>1, and both odd and even d, without separate case tables. In the rectangular representation with d rows and r=(a2+b2)/d columns, the construction uses a constant-time local switch rule, meaning constant time per individual switch, for each q=1,2,…,d−1 at column aq=q−1. Each switch removes two horizontal edges and inserts two vertical edges. The switched horizontal structure forms the first Hamiltonian cycle, while its edge-complement in the Gaussian network forms the second Hamiltonian cycle. Thus, the full edge set is partitioned into two edge-disjoint Hamiltonian cycles. The construction requires O(d) switch-generation time and O(N) time to list the two cycles, where N=a2+b2. Exhaustive validation for all 1≤a≤b≤100, excluding only the degenerate N=2 network, and large-scale validation up to N=3,250,000 confirm implementation correctness and demonstrate practical scalability.