Abstract :
For a positive integer k, a graph G is k-ordered if for every ordered set of k vertices, there is a cycle that encounters the vertices of the set in the given order. If the cycle is also a hamiltonian cycle, then G is said to be k-ordered hamiltonian. This was a concept that was introduced by Ng and Schultz. There has been a series of results involving degree conditions, generalized degree conditions, neighborhood conditions, and forbidden subgraph conditions that imply k-ordered or k-ordered hamiltonian. There have also been results dealing with the same type of conditions for bipartite graphs. Results of this nature will be surveyed.
