Given a graph G. A cycle of G is a sequence of

\r\nvertices of G such that the first and the last vertices are the same.

\r\nA hamiltonian cycle of G is a cycle containing all vertices of G.

\r\nThe graph G is k-ordered (resp. k-ordered hamiltonian) if for any

\r\nsequence of k distinct vertices of G, there exists a cycle (resp.

\r\nhamiltonian cycle) in G containing these k vertices in the specified

\r\norder. Obviously, any cycle in a graph is 1-ordered, 2-ordered and 3-

\r\nordered. Thus the study of any graph being k-ordered (resp. k-ordered

\r\nhamiltonian) always starts with k = 4. Most studies about this topic

\r\nwork on graphs with no real applications. To our knowledge, the

\r\nchordal ring families were the first one utilized as the underlying

\r\ntopology in interconnection networks and shown to be 4-ordered.

\r\nFurthermore, based on our computer experimental results, it was

\r\nconjectured that some of them are 4-ordered hamiltonian. In this

\r\npaper, we intend to give some possible directions in proving the

\r\nconjecture.<\/p>\r\n",
"references": null,
"publisher": "World Academy of Science, Engineering and Technology",
"index": "International Science Index 97, 2015"
}