Implicit Degree Sum Condition for Long Cycles

Abstract

For a vertex $v$ of a graph $G$, Zhu, Li, and Deng introduced the concept of implicit degree $id(v)$, according to the degrees of the neighbors of $v$ and the vertices at distance $2$ with $v$ in $G$. For a subset $S\subseteq V(G)$, let $i{\mathrm{\Delta }}_2(G,S)$ denote the maximum value of the implicit degree sum of two vertices of $S$. In this paper, we will prove: Let $G$ be a $2$-connected graph on $n\ge 3$ vertices and $d$ be a nonnegative integer. If $i{\mathrm{\Delta }}_2(G,S)\ge d$ for each independent set $S$ of order $\kappa (G)+1$, then $G$ has a cycle of length at least $min\{d,n\}$.