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 t 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 m and n with k+ t ≤ m ≤ n, a graph G of order n is (k, t, s,m)-pancyclic if for any (k, t, s)-linear forest F and for each integer r with m ≤ 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 a graph is (k, t, s,m)-pancyclic (or just (k, t,m)-pancyclic) are proved.