Let G be a graph. The partially square graph G~* of G is a graph obtainedfrom G by adding edges uv satisfying the conditions uv E(G), and there is somew ∈N(u)∩N(v), such that N(w) N(u:)∪ N(v)∪ {u, v}. In this paper, we will use thetechnique of the vertex insertion on l-connected (l=k or k+1, k≥2) graphs to providea unified proof for G to be hamiltonian , 1-hamiltonian or hamiltonia11-connected. Thesufficient conditions are expresscd by the inequality concerning sum from i=1 to k |N(Y_i)| and n(Y) in Gfor each independent set Y={y_1, y_2,…,y_k} in G~*, where K_i= {y_i, y_(i-1),…,y_(i-(b-1)) }Y for i ∈{1, 2,…,k} (the subscriptions of y_j's will be taken modulo k), 6 (0
