Pancyclic graphs and linear forests

Given integers k, s, t with 0 <= s <= t and k >= 0, a (k, t, s)-linear forest F is a graph that is the vertex disjoint union of 1 paths with a total of k edges and with s of the paths being single vertices. If the number of single vertex paths is not critical, the forest F will simply be called a (k. t)-linear forest. A graph G of order n >= k + t is (k. t)-hamiltonian if for any (k, t)-linear forest F there is a hamiltonian cycle containing F. More generally, given integers in and it with k + t <= m <= it, a graph G of order it is (k, t, s, in)-pancyclic if for any (k. t, s)-linear forest F and for each integer r with in <= r <= n, there is a cycle of length r containing the linear forest F. Minimum degree conditions and minimum sum of degree conditions of nonadjacent vertices that imply that a graph is (k, t, s, m)-pancyclic (or just (k, t, m)-pancyclic) are proved. (C) 2008 Elsevier B.V. All rights reserved.