An Implicit Degree Condition for Hamiltonian Cycles

Abstract

In [11], Zhu, Li, and Deng introduced the definition of implicit degree of a vertex \(v\), denoted by \(\text{id}(v)\). In this paper, we consider implicit degrees and the hamiltonicity of graphs and obtain that:
If \(G\) is a \(2\)-connected graph of order \(n\) such that \(\text{id}(u) + \text{id}(v) \geq n – 1\) for each pair of vertices \(u\) and \(v\) at distance \(2\), then \(G\) is hamiltonian, with some exceptions.