On k-Ordered Bipartite Graphs

In 1997, Ng and Schultz 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 minimum degree conditions and sum of degree conditions for nonadjacent vertices that imply a balanced bipartite graph to be k-ordered hamiltonian. For example, let G be a balanced bipartite graph on 2n vertices, n sufficiently large. We show that for any positive integer k, if the minimum degree of G is at least (2n+k−1)/4, then G is k-ordered hamiltonian.