Hamiltonian properties of the cube of a 2-edge connected graph

Let G be a 2-edge connected graph with at least 5 vertices. For any given vertices a, b, c, and d in G with a ≠ b, there exists in G₃ a hamiltonian path with endpoints a and b avoiding the edge cd, and there exists in G₃ ∪ {cd} a hamiltonian path with endpoints a and b and containing the edge cd. Also, after removal of two edges or one edge and one vertex from G₃, the resulting graph is still hamiltonian.