A graph G is k-ordered if for any sequence of k distinct vertices of G, there exists a cycle in G containing these k vertices in the specified order. It is k-ordered Hamiltonian if, in addition, the required cycle is Hamiltonian. The question of the existence of an infinite class of 3-regular 4-ordered Hamiltonian graphs was posed in 1997 by Ng and Schultz. At the time, the only known examples were K4 and K3,3. Some progress was made in 2008 by Mészáros, when the Peterson graph was found to be 4-ordered and the Heawood graph was proved to be 4-ordered Hamiltonian; moreover, an infinite class of 3-regular 4-ordered graphs was found. In 2010 a subclass of generalized Petersen graphs was shown to be 4-ordered by Hsu et al., with an infinite subset of this subclass being 4-ordered Hamiltonian, thus answering the open question. In this paper we find another infinite class of 3-regular 4-ordered Hamiltonian graphs, part of a subclass of the chordal ring graphs. In addition, we classify precisely which of these graphs are 4-ordered Hamiltonian.
Bibliographical noteFunding Information:
∗Research partially supported by the NSF-REU Grant DMS 0649099. †Research partially supported by the NSF-REU Grant DMS 0649099. ‡Corresponding author. Research partially supported by the NSF-REU Grant DMS 0649099.
- Chordal ring graphs