A Generalization of Implicit Ore-condition for Hamiltonicity of k-connected Graphs

Abstract

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.