Degree conditions for k-ordered Hamiltonian graphs

For a positive integer k, a graph G is k-ordered hamiltonian if for every ordered sequence of k vertices there is a hamiltonian cycle that encounters the vertices of the sequence in the given order. It is shown that if G is a graph of order n with 3 less than or equal to k less than or equal to n/2, and deg(u) +deg(v) greater than or equal to n+(3k-9)/2 for every pair u,v of nonadjacent vertices of G, then G is k-ordered hamiltonian. Minimum degree conditions are also given for k-ordered hamiltonicity. (C) 2003 Wiley Periodicals, Inc.