On a Hamiltonian Cycle in Which Specified Vertices Are Uniformly Distributed

Let G be a graph with n vertices and minimum degree at least n/2, and 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 path joining them. In this paper, 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.