Classes of Hamilton Cycles in the 5-Cube

A Hamilton cycle in an n-cube is said to be k-warped if its k-paths have their edges running along different parallel 1-factors. No Hamilton cycle in the n-cube can be n-warped. The equivalence classes of Hamilton cycles in the 5-cube are represented by the circuits associated to their corresponding minimum change-number sequences, or minimum H-circuits. This makes feasible an exhaustive search of such Hamilton cycles allowing their classification according to class cardinalities, distribution of change numbers, duplicity, reversibility and k-warped representability, for different values of k < n. This classification boils down to a detailed enumeration of a total of 237675 equivalence classes of Hamilton cycles in the 5-cube, exactly four of which do not traverse any sub-cube. One of these four classes is the unique class of 4-warped Hamilton cycles in the 5-cube. In contrast, there is no 5-warped Hamilton cycle in the 6-cube. On the other hand, there is exactly one class of Hamilton cycles in the graph of middle levels of the 5-cube. A representative of this class possesses an elegant geometrical and symmetrical disposition inside the 5-cube.