On Neighbor-Property Cycles

Cycles of various lengths composed of elements belonging to a set S can be used to define monocyclic permutations (“round-table seatings”) of subsets of S. Those are interesting by themselves, but the whole context becomes still more interesting when the elements possess specific properties or when pairs of elements of S have a specific property (implying a partition of the product set SxS). In this case, for example, one can construct cycles in which every pair of adjacent elements (neighbors) has, or has not, a specific property. For cycle lengths greater than 2, this reduces to finding all distinct Hamiltonian cycles (an NP-complete problem) in a simple graph in which the vertices correspond to the elements of the cycle, and an edge between two vertices is present only if the pair possesses the property.