A Hamiltonian decomposition of a regular graph is a partition of its edge set into Hamiltonian cycles. We consider the second Hamiltonian decomposition problem: for a 4-regular multigraph, find 2 edge-disjoint Hamiltonian cycles different from the given ones. This problem arises in polyhedral combinatorics as a sufficient condition for non-adjacency in the 1-skeleton of the traveling salesperson polytope. We introduce two integer linear programming models for the problem based on the classical Dantzig-Fulkerson-Johnson and Miller-Tucker-Zemlin formulations for the traveling salesperson problem. To enhance the performance on feasible problems, we supplement the algorithm with a variable neighborhood descent heuristic w.r.t. two neighborhood structures and a chain edge fixing procedure. Based on the computational experiments, the Dantzig-Fulkerson-Johnson formulation showed the best results on directed multigraphs, while on undirected multigraphs, the variable neighborhood descent heuristic was especially effective.

正则图的哈密顿分解是将其边集划分为哈密顿循环。我们考虑第二个哈密顿分解问题：对于 4-正则多重图，找到 2 个与给定的不同的边不相交哈密顿循环。这个问题出现在多面体组合学中，作为旅行推销员多面体的 1 骨架中不相邻的充分条件。我们基于旅行推销员问题的经典 Dantzig-Fulkerson-Johnson 和 Miller-Tucker-Zemlin 公式，引入了两个整数线性规划模型。为了提高可行问题的性能，我们用可变邻域下降启发式来补充算法。两个邻域结构和链边缘固定程序。基于计算实验，Dantzig-Fulkerson-Johnson 公式在有向多重图上显示出最佳结果，而在无向多重图上，变量邻域下降启发式尤其有效。