On 4-ordered 3-regular graphs

Abstract

A graph G is k - ordered if for any sequence of k distinct vertices v 1 , v 2 , … , v k 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 4-ordered 3-regular graphs other than the complete graph K 4 and the complete bipartite graph K 3 , 3 . In 2008, Meszaros solved the question by proving that the Petersen graph and the Heawood graph are 4-ordered 3-regular 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 4-ordered 3-regular graphs are vertex transitive. Among these graphs, there are only two non-bipartite graphs, namely the complete graph K 4 and the Petersen graph. In this paper, we prove that there exists a bipartite non-vertex-transitive 4-ordered 3-regular graph of order n for any sufficiently large even integer n . Moreover, there exists a non-bipartite non-vertex-transitive 4-ordered 3-regular graph of order n for any sufficiently large even integer n .