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 $\alpha (G)\le n/2$ such that $\text{id}(v)\ge (n-1)/2$ for each vertex $v$ of $G$, then $G$ is hamiltonian with some exceptions.