Solution to an open problem on 4-ordered Hamiltonian graphs

A graphG is k-ordered if for any sequence of k distinct vertices ofG, 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 [10]. At the time, the only known examples were K4 and K3,3. Some progress was made in 2008 [9] when the Peterson graph was found to be 4-ordered and the Heawood graph was proved to be 4ordered Hamiltonian; moreover an infinite class of 3-regular 4-ordered graphs was found. In this paper we show that a subclass of generalized Petersen graphs are 4-ordered and give a complete classification for which of these graphs are 4-ordered Hamiltonian. In particular, this answers the open question regarding the existence of an infinite class of 3-regular 4ordered Hamiltonian graphs. Moreover, a number of results related to other open problems are presented.