Degree conditions on distance 2 vertices that imply -ordered Hamiltonian

Abstract: For a positive integer , a graph is -ordered if for every ordered set of 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 is said to be -ordered Hamiltonian. We first show that if is a -connected, -ordered graph of order and for every pair of vertices and of with , then is -ordered Hamiltonian unless belongs to an exceptional class of graphs. The latter class is described in this paper. By this result, we prove that is -ordered Hamiltonian if has the order and for every pair of vertices and of with . [Copyright &y& Elsevier]