Essential independent sets and Hamiltonian cycles

An independent set S of a graph G is said to be essential if S has a pair of vertices distance two apart in G. We prove that if every essential independent set S of order k 2 2 in a k-connected graph of order p satisfies max{deg u : u E S} I p, then G is hamiltonian. This generalizes the result of Fan (J. Combinatorial Theory B 37 (19841, 221-227). If w e consider the essential independent sets of order k + 1 instead of k in the assumption of the above statement, we can no longer assure the existence a hamiltonian cycle. However, we can still give a lower bound to the length of a longest cycle.