Precise location of vertices on Hamiltonian cycles

Given k≥2k≥2 fixed positive integers p1,p2,…,pk−1≥2p1,p2,…,pk−1≥2, and kk vertices {x1,x2,…,xk}{x1,x2,…,xk}, let GG be a simple graph of sufficiently large order nn. It is proved that if δ(G)≥(n+2k−2)/2δ(G)≥(n+2k−2)/2, then there is a Hamiltonian cycle CC of GG containing the vertices in order such that the distance along CC is dC(xi,xi+1)=pidC(xi,xi+1)=pi for 1≤i≤k−11≤i≤k−1. Also, let {(xi,yi)|1≤i≤k}{(xi,yi)|1≤i≤k} be a set of kk disjoint pairs of vertices and a graph of sufficiently large graph nn and p1,p2,…,pk≥2p1,p2,…,pk≥2 for k≥2k≥2 fixed positive integers. It will be proved that if δ(G)≥(n+3k−1)/2δ(G)≥(n+3k−1)/2, then there are kk vertex disjoint paths Pi(xi,yi)Pi(xi,yi) of length pipi for 1≤i≤k1≤i≤k.