The neighborhood union of independent sets and hamiltonicity of graphs

Let G   be a graph, N(u)N(u) the neighborhood of uu for each u∈V(G)u∈V(G), and N(U)=⋃u∈UN(u) for each U⊆V(G)U⊆V(G). For any two positive integers s and t  , we prove that there exists a least positive integer N(s,t)N(s,t) such that every (s+t)(s+t)-connected graph G   of order n⩾N(s,t) is hamiltonian if |N(S)|+|N(T)|⩾n|N(S)|+|N(T)|⩾n for every two disjoint independent vertex sets S, T   with |S|=s|S|=s and |T|=t|T|=t.