Tetravalent half-arc-transitive graphs of order $12p$

Document Type : Original paper


1 Department of mathematics, Urmia University, Urmia

2 Department of Mathematics‎, ‎Mazandaran University


A graph is half-arc-transitive if its automorphism group acts transitively on its vertex set, edge set, but not its arc set. In this paper, we study  all tetravalent half-arc-transitive graphs of order $12p$,  where $p$ is a prime.


Main Subjects

[1] B. Alspach and M.Y. Xu, 1/2-transitive graphs of order $3p$, J. Algebraic Combin. 3 (1994), no. 4, 347–355.
[2] I. Antončič and P. Šparl, Classification of quartic half-arc-transitive weak metacirculants of girth at most 4, Discrete Math. 339 (2016), no. 2, 931–945.
[3] R.A. Beezer, Sage for linear algebra a supplement to a first course in linear algebra, Sage web site http://www.sagemath.org (2011).
[4] K. Berčič and M. Ghasemi, Tetravalent arc-transitive graphs of order twice a product of two primes, Discrete Math. 312 (2012), no. 24, 3643–3648.
[5] I.Z. Bouwer, Vertex and edge transitive, but not 1-transitive, graphs, Can. Math. Bull. 13 (1970), no. 2, 231–237.
[6] C.Y. Chao, On the classification of symmetric graphs with a prime number of vertices, Trans. Amer. Math. Soc. 158 (1971), no. 1, 247–256.
[7] H. Cheng and L. Cui, Tetravalent half-arc-transitive graphs of order $p^5$, Appl. Math. Comput. 332 (2018), 506–518.
[8] Y. Cheng and J. Oxley, On weakly symmetric graphs of order twice a prime, J. Combin. Theory, Ser. B 42 (1987), no. 2, 196–211.
[9] M.D.E. Conder and A. Žitnik, Half-arc-transitive graphs of arbitrary even valency greater than 2, European J. Combin. 54 (2016), 177–186.
[10] L. Cui and J.X. Zhou, A classification of tetravalent half-arc-transitive metacirculants of 2-power orders, Appl. Math. Comput. 392 (2021), Article ID: 125755.
[11] S.F. Du and M.Y. Xu, Vertex-primitive 1/2-arc-transitive graphs of smallest order, Commun. Algebra 27 (1999), 163–171.
[12] Y.Q. Feng, J.H. Kwak, X. Wang, and J.X. Zhou, Tetravalent half-arc-transitive graphs of order $2pq$, J. Algebraic Combin. 33 (2011), no. 4, 543–553.
[13] Y.Q. Feng, J.H. Kwak, M.Y. Xu, and J.X. Zhou, Tetravalent half-arc-transitive graphs of order $p^4$, European J. Combin. 29 (2008), no. 3, 555–567.
[14] Y.Q. Feng, J.H. Kwak, and C. Zhou, Constructing even radius tightly attached half-arc-transitive graphs of valency four, J. Algebraic Combin. 26 (2007), no. 4, 431–451.
[15] Y.Q. Feng, K. Wang, and C. Zhou, Tetravalent half-transitive graphs of order $4p$, European J. Combin. 28 (2007), no. 3, 726–733.
[16] A. Gardiner and C.E. Praeger, On 4-valent symmetric graphs, European J. Combin. 15 (1994), no. 4, 375–381.
[17] D. Gorenstein, Finite simple groups, Plenum, New York, 1982.
[18] J.L. Gross and T.W. Tucker, Generating all graph coverings by permutation voltage assignments, Discrete Math. 18 (1977), no. 3, 273–283.
[19] D.F. Holt, A graph which is edge transitive but not arc transitive, J. Graph Theory 5 (1981), no. 2, 201–204.
[20] A. Hujdurovi´c, K. Kutnar, and D. Marušič, Half-arc-transitive group actions with a small number of alternets, J. Combin. Theory, Ser. A 124 (2014), 114–129.
[21] K. Kutnar, D. Marušič, and P. Šparl, An infinite family of half-arc-transitive graphs with universal reachability relation, European J. Combin. 31 (2010), no. 7, 1725–1734.
[22] K. Kutnar, D. Marušič, P. Šparl, R.J. Wang, and M.Y. Xu, Classification of half-arc-transitive graphs of order $4p$, European J. Combin. 34 (2013), no. 7, 1158–1176.
[23] C.H. Li and H.S. Sim, On half-transitive metacirculant graphs of prime-power order, J. Combin. Theory, Ser. B 81 (2001), no. 1, 45–57.
[24] H. Liu, B. Lou, and B. Ling, Tetravalent half-arc-transitive graphs of order $p^2q^2$, Czechoslovak Math. J. 69 (2019), no. 2, 391–401.
[25] D. Marušič, Half-transitive group actions on finite graphs of valency 4, J. Combin. Theory, Ser. B 73 (1998), no. 1, 41–76.
[26] D. Marušič and C.E. Praeger, Tetravalent graphs admitting half-transitive group actions: alternating cycles, J. Combin. Theory, Ser. B 75 (1999), no. 2, 188–205.
[27] P. Potoˇcnik and S. Wilson, A census of edge-transitive tetravalent graphs, Available at http://jan.ucc.nau.edu/~swilson/C4Site/index.html.
[28] C.E. Praeger and M. Xu, A characterisation of a class of symmetric graphs of twice prime valency, European J. Combin. 10 (1986), 91–102.
[29] W.T. Tutte, Connectivity in Graphs, University of Toronto Press, 1966.
[30] R.J. Wang, Half-transitive graphs of order a product of two distinct primes, Commun. Algebra 22 (1994), no. 3, 915–927.
[31] R.J. Wang and M.Y. Xu, A classification of symmetric graphs of order $3p$, J. Combin. Theory, Ser. B 58 (1993), no. 2, 197–216.
[32] X. Wang, Y. Feng, J. Zhou, J. Wang, and Q. Ma, Tetravalent half-arc-transitive graphs of order a product of three primes, Discrete Math. 339 (2016), no. 5, 1566–1573.
[33] X. Wang and Y.Q. Feng, Half-arc-transitive graphs of order $4p$ of valency twice a prime, Ars Math. Contemp. 3 (2010), no. 2, 151–163.
[34] X. Wang and Y.Q. Feng, There exists no tetravalent half-arc-transitive graph of order $2p^2$, Discrete Math. 310 (2010), no. 12, 1721–1724.
[35] X. Wang, J. Wang, and Y. Liu, Tetravalent half-arc-transitive graphs of order $8p$, J. Algebraic Combin. 51 (2019), 237–246.
[36] Y. Wang and Y.Q. Feng, Half-arc-transitive graphs of prime-cube order of small valencies, Ars Math. Contemp. 13 (2017), no. 2, 343–35.
[37] M.Y. Xu, Half-transitive graphs of prime-cube order, J. Algebraic Combin. 1 (1992), no. 3, 275–282.
[38] M.M. Zhang and J.X. Zhou, Tetravalent half-arc-transitive bi-p-metacirculants, J. Graph Theory 92 (2019), no. 1, 19–38.
[39] J.X. Zhou, Tetravalent s-transitive graphs of order $4p$, Discrete Math. 309 (2009), no. 20, 6081–6086.
[40] J.X. Zhou and Y.Q. Feng, Tetravalent one-regular graphs of order $2pq$, J. Algebraic Combin. 29 (2009), 457–471