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 used long node-sequence tables and separate odd/even cases for d. This paper gives a unified closed-form construction that covers both d = 1 and d > 1, and also covers both odd and even d, without separate case tables. In the rectangular representation with d rows and r = (a^2 + b^2)/d columns, the construction uses a constant-time local switch rule 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 = a^2 + b^2. 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 the construction.