On k-ordered Graphs Involved Degree Sum

A graph g is k-ordered Hamiltonian, 2 ≤ k ≤ n, if for every ordered sequence S of k distinct vertices of G, there exists a Hamiltonian cycle that encounters S in the given order. In this article, we prove that if G is a graph on n vertices with degree sum of nonadjacent vertices at least $$n+{{3k - 9} \over 2}$$ , then G is k-ordered Hamiltonian for $$ k = 3,4, \cdots ,{\left\lfloor {\frac{n} {{19}}} \right\rfloor } $$ . We also show that the degree sum bound can be reduced to $$ n + 2{\left\lfloor {\frac{k} {2}} \right\rfloor } - 2 $$ if $$ \kappa {\left( G \right)} \geqslant \frac{{3k - 1}} {2} $$ or δ(G) ≥ 5k − 4. Several known results are generalized.