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 that a graph is ( k , t , s , m ) -pancyclic (or just ( k , t , m ) -pancyclic) are proved.