摘要
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 <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不属于E（G），且J（u,v)≠θ}，这里J（u,v)={ω∈N(u)∩N（V)：N（ω）包含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∑ i=1 ^k|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)}包含Y（yj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}|.
基金
Supported by National Natural Sciences Foundation of China(19971043)
