Insertible vertices, neighborhood intersections, and hamiltonicity

Let G be a simple undirected graph of order n . For an independent set S ⊂ V ( G ) of k vertices, we define the k neighborhood intersections S i = { v ϵ V ( G )\ S | N ( v ) ∩ S| = i }, 1 ≦ i ≦ k , with s i = | S i |. Using the concept of insertible vertices and the concept of neighborhood intersections , we prove the following theorem.