In 2005, Flandrin et al. proved that if G is a k-connected graph of order n and V(G) = X1 ∪X2 ∪ ⋯ UXfc such that d(x) + d(y) ≥ n for each pair of nonadjacent vertices x, y ∈ Xi and each i with i = 1, 2, ⋯, k, then G is hamiltonian. In order to get more sufficient conditions for hamiltonicity of graphs, Zhu, Li and Deng proposed the definitions of two kinds of implicit degree of a vertex v, denoted by id1(v) and id2(v), respectively. In this paper, we are going to prove that if G is a k-connected graph of order n and V(G) = X1 ∪ X2 ∪ ⋯ ∪ Xk such that id2(x) + id2(y) ≥ n for each pair of nonadjacent vertices x, y ∈ Xi and each i with i = 1, 2, ⋯, k, then G is hamiltonian.

中文翻译：

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k-连通图哈密顿性的隐式矿石条件的推广

2005 年，弗兰德林等人。证明了如果 G 是 n 阶 k 连通图且 V(G) = X1 ∪X2 ∪ ⋯ UXfc 使得 d(x) + d(y) ≥ n 对于每对非相邻顶点 x, y ∈ Xi 和每个 i 与 i = 1, 2, ⋯, k，则 G 是哈密顿数。为了得到图的半调性更充分的条件，朱、李和邓提出了顶点v的两种隐式度的定义，分别用id1(v)和id2(v)表示。在本文中，我们将证明，如果 G 是 n 阶 k 连通图且 V(G) = X1 ∪ X2 ∪ ⋯ ∪ Xk 使得 id2(x) + id2(y) ≥ n 对于每一对不相邻的顶点 x, y ∈ Xi 并且每个 i 具有 i = 1, 2, ⋯, k，则 G 是哈密顿的。