Precise location of vertices on Hamiltonian cycles

Given k ≥ 2 fixed positive integers p 1 , p 2 , … , p k − 1 ≥ 2 , and k vertices { x 1 , x 2 , … , x k } , let G be a simple graph of sufficiently large order n . It is proved that if δ ( G ) ≥ ( n + 2 k − 2 ) / 2 , then there is a Hamiltonian cycle C of G containing the vertices in order such that the distance along C is d C ( x i , x i + 1 ) = p i for 1 ≤ i ≤ k − 1 . Also, let { ( x i , y i ) | 1 ≤ i ≤ k } be a set of k disjoint pairs of vertices and a graph of sufficiently large graph n and p 1 , p 2 , … , p k ≥ 2 for k ≥ 2 fixed positive integers. It will be proved that if δ ( G ) ≥ ( n + 3 k − 1 ) / 2 , then there are k vertex disjoint paths P i ( x i , y i ) of length p i for 1 ≤ i ≤ k .