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 \(i\Delta_2(G, S)\) 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 \(n \geq 3\) vertices and \(d\) be a nonnegative integer. If \(i\Delta_2(G, S) \geq d\) for each independent set \(S\) of order \(\kappa(G) + 1\), then \(G\) has a cycle of length at least \(\min\{d, n\}\).