Abstract
This paper contributes to the classification of finite 2-arc-transitive graphs. In [12], all the regular covers of complete bipartite graphs Kn,n were classified, whose covering transformation group is cyclic and whose fibre-preserving automorphism group acts 2-arc-transitively. In this paper, a further classification is achieved for all the regular covers of Kn,n, whose covering transformation group is elementary abelian group of order p2 and whose fibre-preserving automorphism group acts 2-arc-transitively. As a result, two new infinite families of 2-arc-transitive graphs are found. Moveover, it will be explained that it seems to be infeasible to classify all such covers when the covering transformation group is an elementary abelian group of order pk for an arbitrary integer k.
Similar content being viewed by others
References
D. M. Bloom: The subgroups of PSL(3,q) for odd q, Trans. Amer. Math. Soc. 127 (1967), 150–178.
P. J. Cameron: Permutation groups, London Math. Soc. Student Texts 45, Univ. Press, Cambridge, 1999.
M. Conder and J. Ma: Arc-transitive abelian regular covers of cubic graphs, J. Algebra 387 (2013), 215–242.
M. Conder and J. Ma: Arc-transitive abelian regular covers of the Heawood graph, J. Algebra 387 (2013), 243–267.
J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker and R. A. Wilson: Atlas of Finite Groups, Clarendon Press, Oxford, 1985.
J. D. Dixon and B. Mortimer: Permutation groups, Springer-Verlag, New York, 1996.
S. F. Du, D. Marušic and A. O. Waller: On 2-arc-transitive covers of complete graphs, J. Combin. Theory, Ser. B 74 (1998), 276–290.
S. F. Du, J. H. Kwak and R. Nedela: Regular embeddings of complete multipartite graphs, Europ. J. Combin. 26 (2005), 505–519.
S. F. Du, J. H. Kwak and M. Y. Xu: On 2-arc-transitive covers of complete graphs with covering transformation group Z3 p, J. Combin. Theory, Ser. B 93 (2005), 73–93.
S. F. Du, A. Malnic and D. Marušic: Classification of 2-arc-transitive dihedrants, J. Combin. Theory, Ser. B 98 (2008), 1349–1372.
S. F. Du and M. Y. Xu: A classification of semisymmetric graphs of order 2pq, Comm. in Algebra 28 (2000), 2685–2715.
S. F. Du and W. Q. Xu: 2-Arc-transitive cyclic covers of Kn,n, submitted.
S. F. Du and W. Q. Xu: 2-Arc-Transitive regular covers of Kn,n–nK2 having the covering transformation group Z3 p, J. Aust. Math. Soc. 0 (2016), 1–26.
X. G. Fang, G. Havas and C. E. Praeger: On the automorphism groups of quasiprimitive almost simple graphs, J. Algebra 222 (1999), 271–283.
X. G. Fang and C. E. Praeger: Finite two-arc-transitive graphs admitting a Suzuki simple group, Comm. Algebra 27 (1999), 3727–3754.
X. G. Fang and C. E. Praeger: Finite two-arc-transitive graphs admitting a Ree simple group, Comm. Algebra 27 (1999), 3755–3769.
Y. Q. Feng and J. H. Kwak: Cubic symmetric graphs of order a small number times a prime or a prime square, J. Combin. Theory, Ser B 97 (2007), 627–646.
C. D. Godsil and A. D. Hensel: Distance regular covers of the complete graph, J. Combin. Theory, Ser B 56 (1992), 205–238.
C. D. Godsil, R. A. Liebler and C. E. Praeger: Antiposal distance transitive covers of complete graphs, Europ. J. Combin. 19 (1992), 455–478.
A. Gardiner and C. E. Praeger: Topological covers of complete graphs, Math. Proc. Camb. Phil. Soc. 123 (1998), 549–559.
J. L. Gross and T. W. Tucker: Topological Graph Theory, Wiley-Interscience, New York, 1987.
J. L. Gross and T. W. Tucker: Generating all graph coverings by permutation voltage assignments, Discrete Math. 18 (1977), 273–283.
R. M. Guralnick: Subgroups of prime power index in a simple group, J. Algebra 81 (1983), 304–311.
C. Hering: Transitive linear groups and linear groups which contain irreducible subgroups of prime order, Geometriae Dedicata 2 (1974), 425–460.
B. Huppert: Endliche Gruppen I, Springer-Verlag, Berlin, 1967.
A. A. Ivanov and C. E. Praeger: On finite affinne 2-arc-transitive graphs, Europ. J. Combin. 14 (1993), 421–444.
C. H. Li: On finite s-transitive graphs of odd order, J. Combin. Theory, Ser. B 81 (2001), 307–317.
C. H. Li: The finite vertex-primitive and vertex-biprimitive stransitive graphs for s≥4, Tran. Amer. Math. Soc. 353 (2001), 3511–3529.
A. Malnic: Group actions, coverings and lifts of automorphisms, Discrete Math. 182 (1998), 203–218.
B. Mortimer: The modular permutation representations of the known doubly transitive groups, Proc. London Math. Soc. 41 (1980), 1–20.
C. E. Praeger: An O'Nan-Scott theorem for finite quasiprimitive permutation groups and an application to 2-arc transitive graphs, J. London Math. Soc. 47 (1993), 227–239.
C. E. Praeger: On a reduction theorem for finite, bipartite, 2-arc-transitive graphs, Australas J. Combin. 7 (1993), 21–36.
M. Suzuki: Group Theory I, Springer-Verlag, New York, 1982.
W. Q. Xu and S. F. Du: 2-arc-transitive cyclic covers of Kn,n–nK2, J. Algeb. Combin. 39 (2014), 883–902.
W. Q. Xu, S. F. Du, J. H. Kwak and M. Y. Xu: 2-Arc-transitive metacyclic covers of complete graphs, J. Combin. Theory, Ser. B 111 (2015), 54–74.
W. Q. Xu, Y. H. Zhu and S. F. Du: 2-Arc-Transitive regular covers of Kn,n–nK2 with the covering transformation group Z2 p, Ars Math. Contemp. 10 (2016), 269–280.
Author information
Authors and Affiliations
Corresponding author
Additional information
The first two authors are supported by the National Natural Science Foundation of China (11271267, 11371259) and National Research Foundation for the Doctoral Program of Higher Education of China (20121108110005).
The second author is supported by the China Postdoctoral Science Foundation (2016M591271).
The third author is supported by the National Natural Science Foundation of China (11371355).
Rights and permissions
About this article
Cite this article
Du, S., Xu, W. & Yan, G. 2-Arc-Transitive Regular Covers of Kn,n Having the Covering Transformation Group ℤ 2 p . Combinatorica 38, 803–826 (2018). https://doi.org/10.1007/s00493-016-3511-x
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00493-016-3511-x