4‐ordered‐Hamiltonian problems of the generalized Petersen graph GP(n,4)

A graph $G$ is $k$ ‐ordered if for every 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 a Hamiltonian cycle in $G$ . The question of the existence of an infinite class of 3‐regular 4‐ordered‐Hamiltonian graphs was posed in Ng and Schultz in 1997 [2]. At the time, the only known examples of such graphs were $K_4$ and $K_{3,3}$ . Some progress was made by Mészáros in 2008 [21] when the Petersen 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 the generalized Petersen graphs was shown to be 4‐ordered in Hsu et al. [9], with an infinite subset of this subclass being 4‐ordered‐Hamiltonian, thus answering the open question. However, these graphs are bipartite. In this paper we extend the result to another subclass of the generalized Petersen graphs. In particular, we find the first class of infinite non‐bipartite graphs that are both $4$ ‐ordered‐Hamiltonian and $4$ ‐ordered‐Hamiltonian‐connected, which can be seen as a solution to an extension of the question posted in Ng and Schultz in 1997 [2]. (A graph $G$ is $k$ ‐ordered‐Hamiltonian‐connected if for every sequence of $k$ distinct vertices $a_1,a_2,\dots ,a_k$ of $G$ , there exists a Hamiltonian path in $G$ from $a_1$ to $a_k$ where these $k$ vertices appear in the specified order.)