Degree Sum Condition for k-ordered Hamiltonian Connected Graphs

Let G be a graph on n vertices. If for any ordered set of vertices S = {v (1), v (2), . . . , v (k) }, that is, the vertices in S appear in order of the sequence v (1), v (2), . . . , v (k) , there exists a v (1) - v (k) (hamiltonian) path containing S in the given order, then G is k-ordered (hamiltonian) connected. Let {u (1), u (2)} and {u (3), u (4)} be distinct pairs of nonadjacent vertices. When and , we define , otherwise set . In this paper we will present some sufficient conditions for a graph to be k-ordered connected based on . As a main result we will show that if , then G is k-ordered hamiltonian connected. Our outcomes generalize several related results known before.