Classification of the Hamiltonian Cycles in Binary Hypercubes

Abstract

In contrast to the conventional representation of the Hamiltonian cycles in the hypercubes as ring sequences of the numbers of the vertices, the paper proposed to represent the weights of edges connecting the pairs of adjacent cycle vertices as the ring sequences. The weight of an edge is the difference between the numbers of its incident vertices. Relying on the representation of Hamiltonian cycles by sequences of the edge weights, decomposition of the cycle set into classes defined by the distributions of numbers of different edge weights, as well as into kinds belonging to classes and defined by the distributions of edge weights was proposed. It was shown how by the operations of shift and permutation of edge weight one can obtain from the known sequence of edge weights representing some Hamiltonian cycle in the n-dimensional cube at least n!-1 other Hamiltonian cycles of the same class and kind. It was shown how one can pass from the class and kind of the Hamiltonian cycles for the n-dimensional cube to a “similar” class and kind of the Hamiltonian cycles for the (n+1)-dimensional cube.