摘要
Let G be a graph, the square graph G 2 of G is a graph satisfying V(G 2)=V(G) and E(G 2)=E(G)∪｛uv: dist G(u, v)=2｝ . In this paper, we use the technique of vertex insertion on l -connected ( l=k or k+1, k≥2 ) claw-free 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=0N(Y i) and n(Y) in G for each independent set Y=｛y 0, y 1, …, y k｝ of the square graph of G , where b ( 0<b<k+1 ) is an integer, Y i=｛y i, y i-1, …, y i-(b-1)｝Y for i∈｛0, 1, …, k｝ , where subscriptions of y j s will be taken modulo k+1 , and n(Y)=｛v∈ V(G): dist (v, Y)≤ 2｝ .
设G是一个图 ,G的平方图G2 满足V(G2 ) =V(G) ,E(G2 ) =E(G) ∪ {uv :distG(u ,v) =2 } .本文利用插点方法 ,给出了关于k或 (k +1)连通 (k≥ 2 )无爪图G是哈密尔顿的、1 哈密尔顿的或哈密尔顿连通的统一证明 .其充分条件是G中关于∑ki=0N(Yi) 与n(Y)的不等式 ,这里Y={y0 ,y1,… ,yk}是图G2 的任一独立集 ,对于i∈ { 0 ,1,… ,k} ,Yi={ yi,yi- 1,… ,yi- (b- 1) } Y (yj 的下标将取模k+1) ;b是一个整数 ,且 0 <b <k+1;n(Y) ={v∈V(G) :dist(v ,Y)≤ 2 } .
基金
TheNationalNaturalScienceFoundationofChina(No .199710 43 )
