Linear forests and ordered cycles
A collection (1 ≤ t ≤ k) of t disjoint paths, s of them being singletons with |V(L)| = k is called a (k,t,s)-linear forest. A graph G is (k,t,s)-ordered if for every (k,t,s)-linear forest L in G there exists a cycle C in G that contains the paths of L in the designated order as subpaths. If the cycle is also a hamiltonian cycle, then G is said to be (k,t,s)-ordered hamiltonian. We give sharp sum of degree conditions for nonadjacent vertices that imply a graph is (k,t,s)-ordered hamiltonian.