Distributing vertices on Hamiltonian cycles
Let G be a graph of order n and 3?t?n/4 be an integer. Recently, Kaneko and Yoshimoto [J Combin Theory Ser B 81(1) (2001), 100109] provided a sharp ?(G) condition such that for any set X of t vertices, G contains a hamiltonian cycle H so that the distance along H between any two vertices of X is at least n/2t. In this article, minimum degree and connectivity conditions are determined such that for any graph G of sufficiently large order n and for any set of t vertices X?V(G), there is a hamiltonian cycle H so that the distance along H between any two consecutive vertices of X is approximately n/t. Furthermore, the minimum degree threshold is determined for the existence of a hamiltonian cycle H such that the vertices of X appear in a prescribed order at approximately predetermined distances along H.  2011 Wiley Periodicals, Inc. J Graph Theory 69: 2845, 2012
hamiltonian cycles distances