Czechoslovak Mathematical Journal, Vol. 69, No. 2, pp. 391-401, 2019


Tetravalent half-arc-transitive graphs of order $p^2q^2$

Hailin Liu, Bengong Lou, Bo Ling

Received July 15, 2017.   Published online February 4, 2019.

Abstract:  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.
Keywords:  half-arc-transitive graph; normal Cayley graph; cube-free order
Classification MSC:  20B15, 05C25


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

Affiliations:   Hailin Liu, School of Science, Jiangxi University of Science and Technology, No. 86, Hongqi Ave, Ganzhou, Jiangxi 341000, P. R. China, e-mail: hailinliuqp@163.com; Hailin Liu, Bengong Lou (corresponding author), School of Mathematics and Statistics, Yunnan University, No. 2, Cuihubei Rd, Kunming 650091 P. R. China, e-mail: bengong188@163.com; Bo Ling, School of Mathematics and Computer Science, Yunnan Minzu University, No. 2929, Yuehua Ave, Kunming 650504, P. R. China, e-mail: bolinggxu@163.com


 
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