How many Hamiltonian cycles does the cube graph Q3 have?
Should be 3 I would think.
Let G be cubic hamiltonian graph. Then G has precisely three Hamilton cycles if and only if G is a merger of two tups. Moreover, if G has more than two vertices, then G is uniquely 3 -edge-colorable if and only if G is either the merger of a trivial tup and a non-trivial tup that is uniquely 3 -edge-colorable, or else a merger of two non-trivial tups that are uniquely 3 -edge-colorable.