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.
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The authors are very grateful to the anonymous referee for carefully reading the manuscript and providing comments and suggestions which led to a substantial improvement of the paper.
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This paper is supported by the National Natural Science Foundation of China (No.11501322), Scientific Research Foundation for Doctors in Qufu Normal University (No. 2012015) and Natural Science Foundation of Qufu Normal University (No.xkj201415).
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Cai, Jq., Wang, L. A Generalization of Implicit Ore-condition for Hamiltonicity of k-connected Graphs. Acta Math. Appl. Sin. Engl. Ser. 36, 620–626 (2020). https://doi.org/10.1007/s10255-020-0956-x
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DOI: https://doi.org/10.1007/s10255-020-0956-x