On 4-ordered 3-regular graphs

Authors:
Ming Tsai;Tsung-Han Tsai;Jimmy J. M. Tan;Lih-Hsing Hsu
Affiliations:
Department of Computer Science, National Chiao Tung University, Hsinchu, 300, Taiwan ROC;Department of Computer Science, National Chiao Tung University, Hsinchu, 300, Taiwan ROC;Department of Computer Science, National Chiao Tung University, Hsinchu, 300, Taiwan ROC;Department of Computer Science and Information Engineering, Providence University, Taichung, 43301, Taiwan ROC
Venue:
Mathematical and Computer Modelling: An International Journal
Year:
2011

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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.