Let G be a graph. The partially square graph G* of G is a graph obtained from G by adding edges uv satisfying the conditions uv (E) E(G), and there is some w ∈ N(u) ∩ N(v), such that N(w) (U) N(u) ∪ N(v) ∪ {u, v}. In this paper, we will use the technique of the vertex insertion on l-connected (l = k or k + 1, k ≥ 2) graphs to provide a unified proof for G to be hamiltonian, 1-hamiltonian or hamiltonian-connected. The sufficient conditions are expressed by the inequality concerning k∑i=1 |N(Yi)| and n(Y) in G for each independent set Y = {y1, y2,… , yk} in G*, where Yi = {yi, yi-1,… , yi-(b-1)} (U) Y for i ∈ {1, 2,…, k} (the subscriptions of yj's will be taken modulo k), b (0 ＜ b ＜ k) is an integer, and n(Y) = |{v ∈ V(G) : dist(v,Y) ≤ 2}|.%设G是一个图,G的部分平方图G*满足V(G*)=V(G),E(G*)=E(G)∪{uv:uv(E)E(G),且J(u,v)≠()},这里J(u,v)={ω∈N(u)∩N(v):N(ω)(U)N[u]∪N[v]}.本文利用插点方法,给出了关于k或(k+1)-连通(k≥2)图G是哈密尔顿的,1-哈密尔顿的或哈密尔顿连通的统一的证明.其充分条件是G中关于∑|N(Yi)|与n(y)的不等式,这里y={y1,y2,…,yk}是图G*的任一独立集,对于i∈{1,2,…,k},Yi={yi,yi-1,…,yi-(b-1)}(U)y(yi的下标将取模k);b是一个整数,且0＜b＜k;n(y)=|{v∈V(G):dist(v,y)≤2}|.
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