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 $q=1,2,\ldots,d-1$ at column $a_q=q-1$. 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 $1\leq a\leq b\leq 100$, excluding only the degenerate $N=2$ network, and large-scale validation up to $N=3{,}250{,}000$ confirm the construction.

翻译：高斯网络是基于高斯整数剩余类定义的四次对称互连网络。先前工作表明，当生成元 $α=a+bi$ 满足 $\gcd(a,b)=1$ 时，实部与虚部维度可直接构成两个边不交哈密顿环。后续构造将该结果推广至非互质情形 $\gcd(a,b)=d>1$，但其证明需使用长节点序列表，并针对 $d$ 的奇偶性分情况讨论。本文给出统一的闭式构造方法，同时覆盖 $d=1$ 与 $d>1$ 情形，且无需针对 $d$ 的奇偶性分设情况表格。在具有 $d$ 行与 $r=(a^2+b^2)/d$ 列的矩形表示中，该构造对每个 $q=1,2,\ldots,d-1$ 在列 $a_q=q-1$ 处采用常数时间局部切换规则：每次切换移除两条水平边并插入两条垂直边。切换后的水平结构形成第一个哈密顿环，其在高斯网络中的边补结构形成第二个哈密顿环。由此，全部边集被划分为两个边不交的哈密顿环。该构造需 $O(d)$ 的切换生成时间与 $O(N)$ 的环列举时间（其中 $N=a^2+b^2$）。对 $1\leq a\leq b\leq 100$（仅排除退化网络 $N=2$）的穷举验证，以及高达 $N=3{,}250{,}000$ 的大规模验证，均确认了该构造的有效性。