Insertible vertices, neighborhood intersections, and hamiltonicity

Let G be a simple undirected graph of order n. For an independent set S C V(G) of k vertices, we define the k neighborhood intersections S, = { u E V(G)\SIIN(u) n Sl = 11, 1 5 ir k, with s; = ISJ. Using the concept of insertible vertices and the concept of neighborhood intersections, we prove the following theorem. Theorem. Let G be a graph of order n and connectivity K 2 2. Then G is hamiltonian or there exists an independent set X C V(G) of cardinality t + 1, 1 s t 5 K such that