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), 100–109] provided a sharp δ(*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*/2*t*. 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*.

*Journal of Graph Theory*Vol. 69 Iss. 1 (2012) p. 28 - 45 ISSN: 1097-0118

Available at: http://works.bepress.com/colton_magnant/26/