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

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