Hamiltonian cycles of certain kinds of graphs satisfying Dirac condition

Let G be a graph with n vertices and minimum degree at least n/2. Let d be a positive integer such that d ≤ n/4. We define a distance between two vertices as the number of edges of a shortest subpath joining them. In this talk, we show that for any vertex subset A with at most n/2d vertices, there exists a Hamiltonian cycle in which the distance between any two vertices of A is at least d. Furthermore, it is shown that for a set B of vertices with at least 3n/4 vertices, there exists a hamiltonian cycle in which a vertex in B is adjacent to some vertex in B. These results are the best possible.