The spanning diameter of the star graphs

Assume that u and v are any two distinct vertices of different partite sets of S_n with n ≥ 5. We prove that there are (n - 1) internally disjoint paths P₁, P₂, ..., P_n-i joining u to v such that ∪^{n = 1}_{i = 2} P_i spans S_n and l(P_i) ≤ (n - 1)! + 2(n - 2)! + 2(n - 3)! + 1 = n!/(n - 2) + 1. We also prove that there are two internally disjoint paths Q₁ and Q₂ joining u to v such that Q₁ ∪ Q₂ spans S_n and l(Q_i) ≤ n!/2 + l for i = 1,2.