Extension of several sufficient conditions for Hamiltonian graphs

Let G be a 2-connected graph of order n. Suppose that for all 3-independent sets X in G, there exists a vertex u in X such that |N(X∖{u})|+d(u) ≥ n-1. Using the concept of dual closure, we prove that 1. G is hamiltonian if and only if its 0-dual closure is either complete or the cycle C₇ 2. G is nonhamiltonian if and only if its 0-dual closure is either the graph ${\displaystyle (K_r\cup Kₛ\cup Kₜ)\vee K₂}$, 1 ≤ r ≤ s ≤ t or the graph ${\displaystyle \left(\frac{n+1}{2}\right)K₁\vee K_{\frac{n-1}{2}}}$. It follows that it takes a polynomial time to check the hamiltonicity or the nonhamiltonicity of a graph satisfying the above condition. From this main result we derive a large number of extensions of previous sufficient conditions for hamiltonian graphs. All these results are sharp.