EN
An independent set S of a graph G is said to be essential if S has a pair of vertices that are distance two apart in G. In 1994, Song and Zhang proved that if for each independent set S of cardinality k+1, one of the following condition holds:
(i) there exist u ≠ v ∈ S such that d(u) + d(v) ≥ n or |N(u) ∩ N(v)| ≥ α (G);
(ii) for any distinct u and v in S, |N(u) ∪ N(v)| ≥ n - max{d(x): x ∈ S},
then G is Hamiltonian. We prove that if for each essential independent set S of cardinality k+1, one of conditions (i) or (ii) holds, then G is Hamiltonian. A number of known results on Hamiltonian graphs are corollaries of this result.