×

Arc-transitive cyclic covers of graphs with order twice a prime. (English) Zbl 1355.05202

Summary: Arc-transitive cyclic covers of symmetric graphs with a specific prime valency and order twice a prime have been studied by nearly ten papers in the literature. In this paper, we will give a general characterization of arc-transitive cyclic covers of graphs with any prime valency and order twice a prime.

MSC:

05C70 Edge subsets with special properties (factorization, matching, partitioning, covering and packing, etc.)
Full Text: DOI

References:

[1] Cameron, P. J., Finite permutation groups and finite simple groups, Bull. Lond. Math. Soc., 13, 1-22 (1981) · Zbl 0463.20003
[2] Cheng, Y.; Oxley, J., On weakly symmetric graphs of order twice a prime, J. Combin. Theory Ser. B, 42, 196-211 (1987) · Zbl 0583.05032
[3] Du, S. F.; Marušič, D.; Waller, A. O., On 2-arc-transitive covers of complete graphs, J. Combin. Theory Ser. B, 74, 276-290 (1998) · Zbl 1026.05057
[4] Feng, Y. Q.; Kwak, J. H., \(s\)-Regular cyclic coverings of the complete bipartite graphs \(K_{3, 3}\), J. Graph Theory, 45, 101-112 (2004) · Zbl 1033.05053
[5] Feng, Y. Q.; Kwak, J. H., Classifying cubic symmetric graphs of order \(10 p\) or \(10 p^2\), Sci. China, 49, 300-319 (2006) · Zbl 1109.05051
[6] Feng, Y. Q.; Kwak, J. H., Cubic symmetric graphs of order a small number times a prime or a prime square, J. Combin. Theory Ser. B, 97, 627-646 (2007) · Zbl 1118.05043
[7] Feng, Y. Q.; Li, Y. T., One-regular graphs of square-free order of prime valency, European J. Combin., 32, 265-275 (2011) · Zbl 1229.05114
[8] Giudici, M.; Li, C. H.; Praeger, C. E., Analysing finite locally \(s\)-arc-transitive graphs, Trans. Amer. Math. Soc., 356, 291-317 (2003) · Zbl 1022.05033
[9] Godsil, C. D., On the full automorphism group of a graph, Combinatorica, 1, 243-256 (1981) · Zbl 0489.05028
[10] Gorenstein, D., Finite Simple Groups (1982), Plenum Press: Plenum Press New York · Zbl 0182.35402
[11] Gross, J. L.; Tucker, T. W., Generating all graph coverings by permutation voltage assignment, Discrete Math., 18, 273-283 (1977) · Zbl 0375.55001
[12] Huppert, B., Finite Groups (1967), Springer-Verlag: Springer-Verlag Berlin · Zbl 0217.07201
[13] Isaacs, I. M., Finite Group Theory (2008), American Mathematics Society · Zbl 1172.20022
[14] Li, C. H.; Pan, J. M., Finite 2-arc-transitive abelian Cayley graphs, European J. Combin., 29, 148-158 (2008) · Zbl 1193.05089
[15] Malnič, A.; Marušič, D.; Potočnik, P., Elementary abelian covers of graphs, J. Algebraic Combin., 20, 71-97 (2004) · Zbl 1065.05050
[16] Malnič, A.; Marušič, D.; Potočnik, P., On cubic graphs admitting an edge-transitive solvable group, J. Algebraic Combin., 20, 99-113 (2004) · Zbl 1054.05055
[17] Malnič, A.; Nedela, R.; Škoviera, M., Lifting graph automorphisms by voltage assignments, European J. Combin., 21, 927-947 (2000) · Zbl 0966.05042
[18] Malnič, A.; Potočnik, P., Invariant subspaces duality and covers of the Petersen graph, European J. Combin., 27, 971-989 (2006) · Zbl 1091.05033
[19] Pan, J. M., Locally primitive Cayley graphs of dihedral groups, European J. Combin., 36, 39-52 (2014) · Zbl 1284.05129
[20] Pan, J. M.; Huang, Z. H., Arc-transitive regular cyclic covers of the complete bipartite graphs \(K_{p, p}\), J. Algebraic Combin., 42, 619-633 (2015) · Zbl 1319.05069
[21] Pan, J. M.; Huang, Z. H.; Xu, F. H.; Ding, S. Y., On cyclic regular covers of complete graphs of small order, Discrete Math., 331, 36-42 (2014) · Zbl 1297.05193
[22] Praeger, C. E., An O’Nan-Scott theorem for finite quasiprimitive permutation groups and an application to 2-arc transitive graphs, J. Lond. Math. Soc., 47, 227-239 (1992) · Zbl 0738.05046
[23] Schur, I., Untersuchen über die Darstellung der endlichen Gruppen durch gebrochenen linearen Substitutionen, Crelle J., 132, 85-137 (1907) · JFM 38.0174.02
[24] Škoviera, M., A contribution to the theory of voltage graphs, Discrete Math., 61, 281-292 (1986) · Zbl 0594.05029
[25] Wang, C. Q.; Hao, Y., Edge-transitive regular \(Z_n\)-covers of the Heawood graph, Discrete Math., 310, 1752-1758 (2010) · Zbl 1222.05105
[26] Zhou, J. X.; Feng, Y. Q., Edge-transitive dihedral or cyclic covers of cubic symmetric graphs of order \(2 p\), Combinatorica, 34, 1, 115-128 (2014) · Zbl 1313.05175
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.