2-Arc-Transitive Regular Covers of Kn,n Having the Covering Transformation Group ℤp2

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.

[1]  Chris D. Godsil,et al.  Distance regular covers of the complete graph , 1992, J. Comb. Theory, Ser. B.

[2]  Michio Suzuki.  Group Theory I , 1981 .

[3]  Cheryl E. Praeger,et al.  TOPOLOGICAL COVERS OF COMPLETE GRAPHS , 1998 .

[4]  Yanhong Zhu,et al.  2-Arc-Transitive regular covers of K_{n, n} - nK_2 with the covering transformation group Z_p^2 , 2016, Ars Math. Contemp..

[5]  Marston Conder,et al.  Arc-transitive abelian regular covers of the Heawood graph , 2013 .

[6]  P. Cameron,et al.  PERMUTATION GROUPS , 2019, Group Theory for Physicists.

[7]  Peter J. Cameron,et al.  Permutation Groups: Frontmatter , 1999 .

[8]  David M. Bloom,et al.  The subgroups of ${\rm PSL}(3,\,q)$ for odd $q$ , 1967 .

[9]  J. Conway,et al.  ATLAS of Finite Groups , 1985 .

[10]  C. Praeger.  An O'Nan‐Scott Theorem for Finite Quasiprimitive Permutation Groups and an Application to 2‐Arc Transitive Graphs , 1993 .

[11]  Aleksander Malnic,et al.  Group actions, coverings and lifts of automorphisms , 1998, Discret. Math..

[12]  Dragan Marusic,et al.  Classification of 2-arc-transitive dihedrants , 2008, J. Comb. Theory, Ser. B.

[13]  Jonathan L. Gross,et al.  Topological Graph Theory , 1987, Handbook of Graph Theory.

[14]  Cai Heng Li,et al.  On Finite s-Transitive Graphs of Odd Order , 2001, J. Comb. Theory, Ser. B.

[15]  Yan-Quan Feng,et al.  Cubic symmetric graphs of order a small number times a prime or a prime square , 2007, J. Comb. Theory, Ser. B.

[16]  Cheryl E. Praeger,et al.  On a reduction theorem for finite, bipartite 2-arc-transitive graphs , 1993, Australas. J Comb..

[17]  Cheryl E. Praeger,et al.  On Finite Affine 2-Arc Transitive Graphs , 1993, Eur. J. Comb..

[18]  Cheryl E. Praeger,et al.  Antipodal Distance Transitive Covers of Complete Graphs , 1998, Eur. J. Comb..

[19]  R. Guralnick.  Subgroups of prime power index in a simple group , 1983 .

[20]  Cheryl E. Praeger,et al.  Fintte two-arc transitive graphs admitting a suzuki simple group , 1999 .

[21]  Jin Ho Kwak,et al.  Regular embeddings of complete multipartite graphs , 2005, Eur. J. Comb..

[22]  Adrian O. Waller,et al.  On 2-Arc-Transitive Covers of Complete Graphs , 1998, J. Comb. Theory, Ser. B.

[23]  Cheryl E. Praeger,et al.  On the Automorphism Groups of Quasiprimitive Almost Simple Graphs , 1999 .

[24]  Cai Heng Li,et al.  The finite vertex-primitive and vertex-biprimitive -transitive graphs for ≥4 , 2001 .

[25]  Cheryl E. Praeger,et al.  Finite two-are transitive graphs admitting a ree simple group , 1999 .

[26]  Ming Yao Xu,et al.  2-Arc-transitive metacyclic covers of complete graphs , 2015, J. Comb. Theory, Ser. B.

[27]  B. Huppert.  Endliche Gruppen I , 1967 .

[28]  Brain Mortimer,et al.  The Modular Permutation Representations of the Known Doubly Transitive Groups , 1980 .

[29]  Jonathan L. Gross,et al.  Generating all graph coverings by permutation voltage assignments , 1977, Discret. Math..

[30]  J. Dixon,et al.  Permutation Groups , 1996 .

[31]  C. Hering.  Transitive linear groups and linear groups which contain irreducible subgroups of prime order , 1974 .

[32]  Wenqin Xu,et al.  2-Arc-transitive cyclic covers of Kn,n−nK2 , 2014 .

[33]  Marston Conder,et al.  Arc-transitive abelian regular covers of cubic graphs , 2013 .

[34]  Ming-Yao Xu,et al.  A classification of semisymmetric graphs of order 2pq , 2000 .

[35]  Ming Yao Xu,et al.  2-Arc-transitive regular covers of complete graphs having the covering transformation group Z3p , 2005, J. Comb. Theory, Ser. B.