Abstract
We classify tetravalent G-half-arc-transitive graphs Γ of order p2q2, where G ⩽ Aut Γ and p, q are distinct odd primes. This result involves a subclass of tetravalent half-arc-transitive graphs of cube-free order.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
B. Alspach, M. Y. Xu: 1/2-transitive graphs of order 3p. J. Algebr. Comb. 3 (1994), 347–355.
I. Z. Bouwer: Vertex and edge transitive, but not 1-transitive, graphs. Can. Math. Bull. 13 (1970), 231–237.
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.
C. Y. Chao: On the classification of symmetric graphs with a prime number of vertices. Trans. Amer. Math. Soc. 158 (1971), 247–256.
Y. Cheng, J. Oxley: On weakly symmetric graphs of order twice a prime. J. Combin. Theory Ser. B 42 (1987), 196–211.
J. D. Dixon, B. Mortimer: Permutation Groups. Graduate Texts in Mathematics 163, Springer, New York, 1996.
S. F. Du, M. Y. Xu: Vertex-primitive 1/2-arc-transitive graphs of smallest order. Commun. Algebra 27 (1999), 163–171.
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.
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.
C. D. Godsil: On the full automorphism group of a graph. Combinatorica 1 (1981), 243–256.
M. Herzog: On finite simple groups of order divisible by three primes only. J. Algebra 10 (1968), 383–388.
D. F. Holt: A graph which is edge transitive but not arc transitive. J. Graph Theory 5 (1981), 201–204.
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.
B. Huppert: Endliche Gruppen. I. Die Grundlehren der mathematischen Wissenschaften 134, Springer, Berlin, 1967. (In German.)
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.
K. Kutnar, D. Marusic, P. Sparl, R. J. Wang, M. Y. Xu: Classification of half-arc-transitive graphs of order 4p. Eur. J. Comb. 34 (2013), 1158–1176.
C. H. Li: Semiregular automorphisms of cubic vertex transitive graphs. Proc. Am. Math. Soc. 136 (2008), 1905–1910.
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.
C. H. Li, H. S. Sim: On half-transitive metacirculant graphs of prime-power order. J. Comb. Theory Ser. B 81 (2001), 45–57.
B. D. McKay: Transitive graphs with fewer than twenty vertices. Math. Comput. 33 (1979), 1101–1121.
J. Pan, Y. Liu, Z. Huang, C. Liu: Tetravalent edge-transitive graphs of order p 2 q. Sci. China Math. 57 (2014), 293–302.
C. E. Praeger: Finite normal edge-transitive Cayley graphs. Bull. Aust. Math. Soc. 60 (1999), 207–220.
M. Suzuki: Group Theory II. Grundlehren der mathematischen Wissenschaften 248, Springer, New York, 1986.
D. E. Taylor, M. Y. Xu: Vertex-primitive half-transitive graphs. J. Aust. Math. Soc. Ser. A 57 (1994), 113–124.
W. T. Tutte: Connectivity in Graphs. Mathematical Expositions 15, University of Toronto Press, Toronto; Oxford University Press, London, 1966.
R. J. Wang: Half-transitive graphs of order a product of two distinct primes. Commun. Algebra 22 (1994), 915–927.
X. Wang, Y. Q. Feng: Half-arc-transitive graphs of order 4p of valency twice a prime. Ars Math. Contemp. 3 (2010), 151–163.
X. Wang, Y. Q. Feng: There exists no tetravalent half-arc-transitive graph of order 2p 2. Discrete Math. 310 (2010), 1721–1724.
Y. Wang, Y. Q. Feng: Half-arc-transitive graphs of prime-cube order of small valencies. Ars Math. Contemp. 13 (2017), 343–353.
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.
S. Wilson, P. Potočnik: A census of edge-transitive tetravalent graphs, Mini-Census. Available at https://doi.org/www.fmf.uni-lj.si/~potocnik/work.htm.
M. Y. Xu: Half-transitive graphs of prime-cube order. J. Algebr. Comb. 1 (1992), 275–282.
Author information
Authors and Affiliations
Corresponding author
Additional information
This work was partially supported by the NNSF of China (11861076, 11701503, 11761079), the Science and Technology Research Project of Jiangxi Education Department (GJJ180488), the Doctoral Fund Project of Jiangxi University of Science and Technology (jxxjbs18035), and the NSF of Yunnan Province (2018FB003).
Rights and permissions
About this article
Cite this article
Liu, H., Lou, B. & Ling, B. Tetravalent half-arc-transitive graphs of order p2q2. Czech Math J 69, 391–401 (2019). https://doi.org/10.21136/CMJ.2019.0335-17
Received:
Published:
Issue Date:
DOI: https://doi.org/10.21136/CMJ.2019.0335-17