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 \(i\Delta_2(S)\) 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 \(n \geq 3\) vertices. If \(i\Delta_2(S) \geq d\) 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., \(308 (2008) 6584-6587]\).