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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: 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. 2010 Mathematics Subject Classification: 05C38, 05C45.doi:10.4064/cm120-1-5 fatcat:hqt5f5ygzbfjnmxib6cuar3hfy