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One-regular cubic graphs of order a small number times a prime or a prime square

Part of: Graph theory Permutation groups

Published online by Cambridge University Press:  09 April 2009

Yan-Quan Feng  and
Jin Ho Kwak
Yan-Quan Feng
Affiliation:
Department of Mathematics, Beijing Jiaotong University, Beijing 100044, P.R. China, e-mail: [email protected]
Jin Ho Kwak
Affiliation:
Department of Mathematics, Pohang University of Science, and Technology, Pohang, 790–784 Korea e-mail: [email protected]
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Abstract

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A graph is one-regular if its automorphism group acts regularly on the set of its arcs. In this paper we show that there exists a one-regular cubic graph of order 2p or 2p2 where p is a prime if and only if 3 is a divisor of p – 1 and the graph has order greater than 25. All of those one-regular cubic graphs are Cayley graphs on dihedral groups and there is only one such graph for each fixed order. Surprisingly, it can be shown that there is no one-regular cubic graph of order 4p or 4p2.

Keywords

Cayley grapharc-transitive graphone-regular graph

MSC classification

Secondary: 05C25: Graphs and abstract algebra (groups, rings, fields, etc.) 20B25: Finite automorphism groups of algebraic, geometric, or combinatorial structures
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 76 , Issue 3 , June 2004 , pp. 345 - 356
Copyright
Copyright © Australian Mathematical Society 2004

References

[1]Alspach, B., Marušič, D. and Nowitz, L., ‘Constructing graphs which are ½-transitive’, J. Austral. Math. Soc. (A) 56 (1994), 391–402.CrossRefGoogle Scholar
[2]Cheng, Y. and Oxley, J., ‘On weakly symmetric graphs of order twice a prime’, J. Combin. Theory (B) 42 (1987), 196–211.CrossRefGoogle Scholar
[3]Conder, M. D. E. and Dobcsányi, P., ‘Trivalent symmetric graphs on up to 768 vertices’, J. Combin. Math. Combin. Comput. 40 (2002), 41–63.Google Scholar
[4]Conder, M. D. E. and Praeger, C. E., ‘Remarks on path-transitivity on finite graphs’, Europ. J. Combin. 17 (1996), 371–378.CrossRefGoogle Scholar
[5]Conway, J. H., Curtis, R. T., Norton, S. P., Parker, R. A. and Wilson, R. A., Atlas of finite groups (Oxford University Press, Oxford, 1985).Google Scholar
[6]Djoković, D. Ž. and Miller, G. L., ‘Regular groups of automorphisms of cubic graphs’, J. Combin. Theory (B) 29 (1980), 195–230.CrossRefGoogle Scholar
[7]Fang, X. G., Wang, J. and Xu, M. Y., ‘On l-arc-regular graphs’, Europ. J. Combin. 23 (2002), 785–791.CrossRefGoogle Scholar
[8]Feng, Y. Q., Kwak, J. H. and Xu, M. Y., ‘s-regular cubic Cayley graphs on abelian or dihedral groups’, Research Report No. 53, (Institute of Math. and School of Math. Sci., Peking Univ., 2000).Google Scholar
[9]Frucht, R., ‘A one-regular graph of degree three’, Canad. J. Math. 4 (1952), 240–247.CrossRefGoogle Scholar
[10]Gorenstein, D., Finite simple groups (Plenum Press, New York, 1982).CrossRefGoogle Scholar
[11]Gross, J. L. and Tucker, T. W., ‘Generating all graph converings by permutation voltage assignment’, Discrete Math. 18 (1977), 273–283.CrossRefGoogle Scholar
[12]Harary, F., Graph theory (Addison-Wesley, Reading, MA, 1969).CrossRefGoogle Scholar
[13]Huppert, B., Endliche Gruppen I (Springer, Berlin, 1967).CrossRefGoogle Scholar
[14]Lorimer, P., ‘Vertex-transitive graphs: symmetric graphs of prime valency’, J. Graph Theory 8 (1984), 55–68.CrossRefGoogle Scholar
[15]Malnič, A., Marušič, D. and Seifter, N., ‘Constructing infinite one-regular graphs’, Europ. J. Combin. 20 (1999), 845–853.CrossRefGoogle Scholar
[16]Malnič, A., Nedela, R. and Škoviera, M., ‘Lifting graph automorphisms by voltage assignments’, Europ. J. Combin. 21 (2000), 924–947.CrossRefGoogle Scholar
[17]Marušič, D., ‘A family of one-regular graphs of valency 4’, Europ. J. Combin. 18 (1997), 59–64.CrossRefGoogle Scholar
[18]Marušič, D. and Nedela, R., ‘Maps and half-transitive graphs of valency 4’, Europ. J. Combin. 19 (1998), 345–354.CrossRefGoogle Scholar
[19]Marušič, D. and Xu, M. Y., ‘A ½-transitive graph of valency 4 with a nonsolvable group of automorphisms’, J. Graph Theory 25 (1997), 133–138.3.0.CO;2-N>CrossRefGoogle Scholar
[20]Miller, R. C., ‘The trivalent symmetric graphs of girth at most six’, J. Combin. Theory (B) 10 (1971), 163–182.CrossRefGoogle Scholar
[21]Praeger, C. E., ‘Imprimitive symmetric graphs’, Ars Combinatoria 19A (1985), 149–163.Google Scholar
[22]Robinson, D. J., A course in the theory of groups (Springer, New York, 1982).CrossRefGoogle Scholar
[23]Wielandt, H., Finite permutation groups (Academic Press, New York, 1964).Google Scholar
[24]Xu, M. Y., ‘A note on one-regular graphs’, Chinese Sci. Bull. 45 (2000), 2160–2162.Google Scholar