An implicit degree condition for k-connected 2-heavy graphs to be hamiltonian

Let id(v) denote the implicit degree of a vertex v in a graph G. An independent set S of G is said to be essential if S contains a pair of vertices at distance 2 in G. A graph G on n≥3 vertices is called 2-heavy if there exist at least two end-vertices of every induced claw having implicit degree at least n/2. In this paper, we prove that: Let G be a k-connected (k≥2) 2-heavy graph on n≥3 vertices. If max⁡{id(v):v∈S}≥n/2 for every essential independent set S of order k, then G is hamiltonian.