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.