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 $\text{id}(u)+\text{id}(v)\ge n–1$ for each pair of vertices $u$ and $v$ at distance $2$, then $G$ is hamiltonian, with some exceptions.