An Implicit Degree Dirac Condition for Hamiltonian Cycles

Abstract

In \(1989\), Zhu, Li, and Deng introduced the definition of implicit degree, denoted by \(\text{id}(v)\), of a vertex \(v\) in a graph \(G\) and they obtained sufficient conditions for a graph to be hamiltonian with the implicit degrees. In this paper, we prove that if \(G\) is a \(2\)-connected graph of order \(n\) with \(\alpha(G) \leq n/2\) such that \(\text{id}(v) \geq (n-1)/2\) for each vertex \(v\) of \(G\), then \(G\) is hamiltonian with some exceptions.