Long cycles containing k-ordered vertices in graphs

Let G be a (k + 2)-connected graph on n vertices and S = {nu(1), nu(2),...,nu(k)} be any ordered set of vertices, that is, the vertices in S appear in the order of the sequence nu(1), nu(2),...,nu(k). We will show that if there exists a cycle containing S in the given order, then there exists a cycle C containing S in the given order such that vertical bar C vertical bar >= min{n, sigma(2)(G)} where sigma(2)(G) = min{d(G)(u) + d(G)(nu): u, nu E V(G); u nu is not an element of E(G)} when G is not complete, otherwise set sigma(2)(G) = infinity. This generalizes several related results known before. (c) 2007 Elsevier B.V. All rights reserved.