Mutually independent Hamiltonian cycles in hypercubes

A Hamiltonian cycle C of G is described as 1, u₂, ..., u_n(G), u₁> to emphasize the order of nodes in C. Thus, u₁ is the beginning node and u_i is the i-th node in C. Two Hamiltonian cycles of G beginning at u, C₁=1, v₂, ..., v_n(G), v₁> and C₂=1, u₂, ..., u_n(G), u₁>, are independent if u=v₁=u₁, and v_i≠u_i for 11, C₂, ..., C_k} of G are mutually independent if any two different Hamiltonian cycles are independent. The mutually independent Hamiltonianicity of graph G, IHC(G), is the maximum integer k such that for any node u of G there exist k-mutually independent Hamiltonian cycles of G starting at u. Let Q_n be the n-dimensional hypercube. We prove that IHC(Q_n)=n-1 if n∈{1, 2, 3} and IHC(Q_n)=n if n≥4.