**Abstract:** |
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 specied order.
It is k-ordered Hamiltonian if, in addition, the required cycle is Hamiltonian. The question of the existence of an innite class of 3-regular 4-ordered
Hamiltonian graphs was posed in 1997. At the time, the only known examples were K4 and K3;3. Some progress was made in 2008 when the
Peterson graph was found to be 4-ordered and the Heawood graph was
proved to be 4-ordered Hamiltonian; moreover an innite 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 classication
for which of these graphs are 4-ordered Hamiltonian. In particular, this
answers the open question regarding the existence of an innite class of
3-regular 4-ordered Hamiltonian graphs. Moreover, a number of results
related to other open problems are presented. |
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