| Title:
|
Tetravalent half-arc-transitive graphs of order $p^2q^2$ (English) |
| Author:
|
Liu, Hailin |
| Author:
|
Lou, Bengong |
| Author:
|
Ling, Bo |
| Language:
|
English |
| Journal:
|
Czechoslovak Mathematical Journal |
| ISSN:
|
0011-4642 (print) |
| ISSN:
|
1572-9141 (online) |
| Volume:
|
69 |
| Issue:
|
2 |
| Year:
|
2019 |
| Pages:
|
391-401 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
We classify tetravalent $G$-half-arc-transitive graphs $\Gamma $ of order $p^2q^2$, where $G\leq \mathop {\textsf {Aut}}\Gamma $ and $p$, $q$ are distinct odd primes. This result involves a subclass of tetravalent half-arc-transitive graphs of cube-free order. (English) |
| Keyword:
|
half-arc-transitive graph |
| Keyword:
|
normal Cayley graph |
| Keyword:
|
cube-free order |
| MSC:
|
05C25 |
| MSC:
|
20B15 |
| idZBL:
|
Zbl 07088792 |
| idMR:
|
MR3959952 |
| DOI:
|
10.21136/CMJ.2019.0335-17 |
| . |
| Date available:
|
2019-05-24T08:57:06Z |
| Last updated:
|
2021-07-05 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/147732 |
| . |
| Reference:
|
[1] Alspach, B., Xu, M. Y.: 1/2-transitive graphs of order $3p$.J. Algebr. Comb. 3 (1994), 347-355. Zbl 0808.05056, MR 1293821, 10.1023/A:1022466626755 |
| Reference:
|
[2] Bouwer, I. Z.: Vertex and edge transitive, but not 1-transitive, graphs.Can. Math. Bull. 13 (1970), 231-237. Zbl 0205.54601, MR 0269532, 10.4153/CMB-1970-047-8 |
| Reference:
|
[3] Bray, J. N., Holt, D. F., Roney-Dougal, C. M.: The Maximal Subgroups of the Low-Dimensional Finite Classical Groups.London Mathematical Society Lecture Note Series 407, Cambridge University Press, Cambridge (2013). Zbl 1303.20053, MR 3098485, 10.1017/CBO9781139192576 |
| Reference:
|
[4] Chao, C. Y.: On the classification of symmetric graphs with a prime number of vertices.Trans. Amer. Math. Soc. 158 (1971), 247-256. Zbl 0217.02403, MR 0279000, 10.2307/1995785 |
| Reference:
|
[5] Cheng, Y., Oxley, J.: On weakly symmetric graphs of order twice a prime.J. Combin. Theory Ser. B 42 (1987), 196-211. Zbl 0583.05032, MR 0884254, 10.1016/0095-8956(87)90040-2 |
| Reference:
|
[6] Dixon, J. D., Mortimer, B.: Permutation Groups.Graduate Texts in Mathematics 163 Springer, New York (1996). Zbl 0951.20001, MR 1409812, 10.1007/978-1-4612-0731-3 |
| Reference:
|
[7] Du, S. F., Xu, M. Y.: Vertex-primitive 1/2-arc-transitive graphs of smallest order.Commun. Algebra 27 (1999), 163-171. Zbl 0922.05032, MR 1668232, 10.1080/00927879908826426 |
| Reference:
|
[8] Feng, Y. Q., Kwak, J. H., Wang, X., Zhou, J. X.: Tetravalent half-arc-transitive graphs of order {$2pq$}.J. Algebr. Comb. 33 (2011), 543-553. Zbl 1226.05134, MR 2781962, 10.1007/s10801-010-0257-1 |
| Reference:
|
[9] Feng, Y. Q., Kwak, J. H., Xu, M. Y., Zhou, J. X.: Tetravalent half-arc-transitive graphs of order {$p^4$}.Eur. J. Comb. 29 (2008), 555-567. Zbl 1159.05024, MR 2397337, 10.1016/j.ejc.2007.05.004 |
| Reference:
|
[10] Godsil, C. D.: On the full automorphism group of a graph.Combinatorica 1 (1981), 243-256. Zbl 0489.05028, MR 0637829, 10.1007/BF02579330 |
| Reference:
|
[11] Herzog, M.: On finite simple groups of order divisible by three primes only.J. Algebra 10 (1968), 383-388. Zbl 0167.29101, MR 0233881, 10.1016/0021-8693(68)90088-4 |
| Reference:
|
[12] Holt, D. F.: A graph which is edge transitive but not arc transitive.J. Graph Theory 5 (1981), 201-204. Zbl 0423.05020, MR 0615008, 10.1002/jgt.3190050210 |
| Reference:
|
[13] Hujdurović, A., Kutnar, K., Marušič, D.: Half-arc-transitive group actions with a small number of alternets.J. Comb. Theory, Ser. A 124 (2014), 114-129. Zbl 1283.05126, MR 3176193, 10.1016/j.jcta.2014.01.005 |
| Reference:
|
[14] Huppert, B.: Endliche Gruppen. I.Die Grundlehren der mathematischen Wissenschaften 134, Springer, Berlin German (1967). Zbl 0217.07201, MR 0224703, 10.1007/978-3-642-64981-3 |
| Reference:
|
[15] Huppert, B., Lempken, W.: Simple groups of order divisible by at most four primes.Izv. Gomel. Gos. Univ. Im. F. Skoriny 16 (2000), 64-75. Zbl 1159.20303 |
| Reference:
|
[16] Kutnar, K., Marušič, D., Šparl, P., Wang, R. J., Xu, M. Y.: Classification of half-arc-transitive graphs of order $4p$.Eur. J. Comb. 34 (2013), 1158-1176. Zbl 1292.05134, MR 3055230, 10.1016/j.ejc.2013.04.004 |
| Reference:
|
[17] Li, C. H.: Semiregular automorphisms of cubic vertex transitive graphs.Proc. Am. Math. Soc. 136 (2008), 1905-1910. Zbl 1157.05028, MR 2383495, 10.1090/S0002-9939-08-09217-4 |
| Reference:
|
[18] Li, C. H., Lu, Z. P., Zhang, H.: Tetravalent edge-transitive Cayley graphs with odd number of vertices.J. Comb. Theory. Ser. B 96 (2006), 164-181. Zbl 1078.05039, MR 2185986, 10.1016/j.jctb.2005.07.003 |
| Reference:
|
[19] Li, C. H., Sim, H. S.: On half-transitive metacirculant graphs of prime-power order.J. Comb. Theory Ser. B 81 (2001), 45-57. Zbl 1024.05038, MR 1809425, 10.1006/jctb.2000.1992 |
| Reference:
|
[20] McKay, B. D.: Transitive graphs with fewer than twenty vertices.Math. Comput. 33 (1979), 1101-1121. Zbl 0411.05046, MR 0528064, 10.2307/2006085 |
| Reference:
|
[21] Pan, J., Liu, Y., Huang, Z., Liu, C.: Tetravalent edge-transitive graphs of order {$p^2q$}.Sci. China Math. 57 (2014), 293-302. Zbl 1286.05071, MR 3150279, 10.1007/s11425-013-4708-8 |
| Reference:
|
[22] Praeger, C. E.: Finite normal edge-transitive Cayley graphs.Bull. Aust. Math. Soc. 60 (1999), 207-220. Zbl 0939.05047, MR 1711938, 10.1017/S0004972700036340 |
| Reference:
|
[23] Suzuki, M.: Group Theory II.Grundlehren der mathematischen Wissenschaften 248, Springer, New York (1986). Zbl 0586.20001, MR 0815926 |
| Reference:
|
[24] Taylor, D. E., Xu, M. Y.: Vertex-primitive half-transitive graphs.J. Aust. Math. Soc. Ser. A 57 (1994), 113-124. Zbl 0808.05055, MR 1279290, 10.1017/S1446788700036090 |
| Reference:
|
[25] Tutte, W. T.: Connectivity in Graphs.Mathematical Expositions 15, University of Toronto Press, Toronto; Oxford University Press, London (1966). Zbl 0146.45603, MR 0210617 |
| Reference:
|
[26] Wang, R. J.: Half-transitive graphs of order a product of two distinct primes.Commun. Algebra 22 (1994), 915-927. Zbl 0795.05072, MR 1261014, 10.1080/00927879408824885 |
| Reference:
|
[27] Wang, X., Feng, Y. Q.: Half-arc-transitive graphs of order {$4p$} of valency twice a prime.Ars Math. Contemp. 3 (2010), 151-163. Zbl 1213.05129, MR 2729365, 10.26493/1855-3974.125.164 |
| Reference:
|
[28] Wang, X., Feng, Y. Q.: There exists no tetravalent half-arc-transitive graph of order $2p^2$.Discrete Math. 310 (2010), 1721-1724. Zbl 1223.05119, MR 2610274, 10.1016/j.disc.2009.11.020 |
| Reference:
|
[29] Wang, Y., Feng, Y. Q.: Half-arc-transitive graphs of prime-cube order of small valencies.Ars Math. Contemp. 13 (2017), 343-353. Zbl 1380.05042, MR 3720537, 10.26493/1855-3974.964.594 |
| Reference:
|
[30] Wang, X., Feng, Y., Zhou, J., Wang, J., Ma, Q.: Tetravalent half-arc-transitive graphs of order a product of three primes.Discrete Math. 339 (2016), 1566-1573. Zbl 1333.05144, MR 3475570, 10.1016/j.disc.2015.12.022 |
| Reference:
|
[31] Wilson, S., Potočnik, P.: A census of edge-transitive tetravalent graphs, Mini-Census.Available at https://www.fmf.uni-lj.si/ {potocnik/work.htm}. |
| Reference:
|
[32] Xu, M. Y.: Half-transitive graphs of prime-cube order.J. Algebr. Comb. 1 (1992), 275-282. Zbl A0786.05044, MR 1194079, 10.1023/A:1022440002282 |
| . |
