A hamiltonian cycle C of a graph G is described as ?u1, u2, . . . , un(G), u1? to emphasize the order of vertices in C. Thus, u1 is the start vertex and ui is the i-th vertex in C. Two hamiltonian cycles of G start at a vertex x, C1 = ?u1, u2, . . . , un(G), u1? and C2 = ?v1, v2, . . . , vn(G), v1?, are independent if x = u1 = v1 and ui ? vi for every i, 2 ≤ i ≤ n(G). A set of hamiltonian cycles {C1, C2, . . . , Ck} of G are mutually independent if any two different hamiltonian cycles are independent. The mutually independent hamiltonicity of graph G, IHC(G), is the maximum integer k such that for any vertex u of G there exist k-mutually independent hamiltonian cycles of G starting at u. In this paper, we are going to study IHC(G) for the n-dimensional pancake graph Pn and the n-dimensional star graph Sn. We prove that IHC(Pn) = n − 1 if n ≥ 4 and IHC(Sn) = n − 1 if n ≥ 5.