Some 3 - regular 4 - ordered graphs ∗

A graph G is k-ordered if for any sequence of k distinct vertices v1, v2, . . . , vk of G there exists a cycle in G containing these k vertices in the specified order. In 1997, Ng and Schultz posed the question of the existence of 3-regular 4-ordered graphs other than K4 and K3,3. In 2008, Meszaros solve the question by proving the Petersen graph and the Heawood graph are 3-regular 4-ordered graphs. Moreover, the generalized Honeycomb torus GHT(3, n, 1) is 4-ordered for any even integer n with n ≥ 8. Up to now, all the known 3-regular 4-ordered graphs are vertex transitive. Among these graphs, there are only two nonbipartite graphs, namely the complete graph K4 and the Petersen graph. In this paper, we prove that there exists a bipartite non-vertex-transitive 4-ordered 3-regular graphs of order n for any sufficient large even integer n.