An implicit degree condition for long cycles in 2-connected graphs

Let id (v) denote the implicit degree of a vertex v. In this work we prove that: If G is a 2-connected graph with max {id (u), id (v)} ≥ c / 2 for each pair of nonadjacent vertices u and v that are vertices of an induced claw or an induced modified claw of G, then G contains either a Hamilton cycle or a cycle of length at least c. This extends several previous results on the existence of long cycles in graphs.