A Generalization of Implicit Ore-condition for Hamiltonicity of K-Connected Graphs

In 2005, Flandrin et al. proved that if G is a k -connected graph of order n and V ( G ) = X 1 ∪ X 2 ∪ ⋯ UXfc such that d ( x ) + d ( y ) ≥ n for each pair of nonadjacent vertices x, y ∈ X i 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 id 1 ( v ) and id 2 ( v ), respectively. In this paper, we are going to prove that if G is a k -connected graph of order n and V ( G ) = X 1 ∪ X 2 ∪ ⋯ ∪ X k such that id 2 ( x ) + id 2 ( y ) ≥ n for each pair of nonadjacent vertices x, y ∈ X i and each i with i = 1, 2, ⋯, k , then G is hamiltonian.