Abstract: Given a graph G. A cycle of G is a sequence of
vertices of G such that the first and the last vertices are the same.
A hamiltonian cycle of G is a cycle containing all vertices of G.
The graph G is k-ordered (resp. k-ordered hamiltonian) if for any
sequence of k distinct vertices of G, there exists a cycle (resp.
hamiltonian cycle) in G containing these k vertices in the specified
order. Obviously, any cycle in a graph is 1-ordered, 2-ordered and 3-
ordered. Thus the study of any graph being k-ordered (resp. k-ordered
hamiltonian) always starts with k = 4. Most studies about this topic
work on graphs with no real applications. To our knowledge, the
chordal ring families were the first one utilized as the underlying
topology in interconnection networks and shown to be 4-ordered.
Furthermore, based on our computer experimental results, it was
conjectured that some of them are 4-ordered hamiltonian. In this
paper, we intend to give some possible directions in proving the
conjecture.