On k-ordered graphs

Ng and Schultz [J Graph Theory 1 (1997), 45±57] introduced the idea of cycle orderability. For a positive integer k, a graph G is k-ordered if for every ordered sequence of k vertices, there is a cycle that encounters the vertices of the sequence in the given order. If the cycle is also a Hamiltonian cycle, then G is said to be k-ordered Hamiltonian. We give sum of degree conditions for nonadjacent vertices and neighborhood union conditions that imply a graph is k-ordered Hamiltonian. ß 2000 John Wiley & Sons, Inc. J Graph Theory 35: 69±82, 2000