Essential Independent Condition for Graphs to Be Hamiltonian

Let G be a graph of order n. For graph to be Hamiltonian, beginning with Dirac's classic result in 1952, Dirac's theorem was followed by that of Ore in 1960. In 1984, Fan generalized Dirac's theorem and Ore's theorem as if G is a 2-connected graph of order n and max {d (u),d (v)}≥n/2 for each pair of vertices u and v with d (u,v)=2, then G is hamiltonian. In 1991, Faudree et al proved that if G is a 2-connected graph and, |N (u)∪N (v)|+δ(G)≥n for each pair of nonadjacent vertices u,v∈V(G), then G is hamiltonian. This paper generalizes the above conditions of Dirac, Ore, Fan and Faudree et al in the case of 3-connected graph and proves that if G is a 3-connected graph of order n and max{|N(x)∪ N (y)| +d (u), |N (w)∪N (z)|+d (v)}≥n for every choice of 6 Essential independent vertices, then G is hamiltonian.