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We consider a Hamiltonian decomposition problem of partitioning a regulargraph into edge-disjoint Hamiltonian cycles. A sufficient condition for vertexadjacency in the 1-skeleton of the traveling salesperson polytope can beformulated as the Hamiltonian decomposition problem in a 4-regular multigraph.We introduce a heuristic general variable neighborhood search algorithm forthis problem based on finding a vertex-disjoint cycle cover of the multigraphthrough reduction to perfect matching and several cycle merging operations. Thealgorithm has a one-sided error: the answer "not adjacent" is always correct,and was tested on random directed and undirected Hamiltonian cycles and onpyramidal tours.

Hamiltonian decomposition and verifying vertex adjacency in 1-skeleton of the traveling salesperson polytope by variable neighborhood search