On 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 its neighbors and the vertices at distance $2$ with $v$ in $G$. For $S\subset V(G)$, let $i{\mathrm{\Delta }}_2(S)$ denote the maximum value of the implicit degree sum of two vertices of $S$. In this paper, we will prove the following result: Let $G$ be a $2$-connected graph on $n\ge 3$ vertices. If $i{\mathrm{\Delta }}_2(S)\ge d$ for each independent set $S$ of order $\kappa (G)+1$, then $G$ has a cycle of length at least $min\{d,n\}$. This result generalizes one result of Yamashita [T. Yamashita, On degree sum conditions for long cycles and cycles through specified vertices, Discrete Math., $308(2008)6584-6587]$.