Advertisement

No hexavalent half-arc-transitive graphs of order twice a prime square exist

  • Mi-Mi Zhang
Article
  • 46 Downloads

Abstract

A graph is half-arc-transitive if its automorphism group acts transitively on its vertex set and edge set, but not arc set. Let p be a prime. Wang and Feng (Discrete Math. 310 (2010) 1721–1724) proved that there exists no tetravalent half-arc-transitive graphs of order \(2p^2\). In this paper, we extend this result to prove that no hexavalent half-arc-transitive graphs of order \(2p^2\) exist.

Keywords

Half-arc-transitive bi-Cayley graph vertex transitive edge transitive 

2010 Mathematics Subject Classification

05C25 20B25 

References

  1. 1.
    Alspach B, Marušič D and Nowitz L, Constructing graphs which are 1/2-transitive, J. Aust. Math. Soc. A. 56 (1994) 391–402MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Bouwer I Z, Vertex and edge-transitive but not 1-transitive graphs, Can. Math. Bull. 13 (1970) 231–237MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Bosma W, Cannon J and Playoust C, The MAGMA algebra system I: the user language, J. Symb. Comput. 24 (1997) 235–265MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Bondy J A and Murty U S R, Graph theory with applications (1976) (New York: Elsevier North Holland)CrossRefzbMATHGoogle Scholar
  5. 5.
    Chao C Y, On the classification of symmetric graphs with a prime number of vertices, Trans. Am. Math. Soc. 158 (1971) 247–256MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Conway H J, Curtis R T, Norton S P, Parker R A and Wilson R A, Atlas of finite group (1985) (Oxford: Oxford University Press) pp. 9–13zbMATHGoogle Scholar
  7. 7.
    Conder M D E and Marušič D, A tetravalent half-arc-transitive graph with non-abelian vertex stabilizer, J. Comb. Theory B. 88 (2003) 67–76MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Cheng Y and Oxley J, On weakly symmetric graphs of order twice a prime, J. Comb. Theory B. 42 (1987) 196–211MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Conder M D E, Zhou J X, Feng Y Q and Zhang M M, Finite normal edge-transitive bi-Cayley graphs, arXiv:1606.04625 [math.CO]
  10. 10.
    Doyle P G, On transitive graphs, Senior Thesis (1976) (Cambridge: Harvard College)Google Scholar
  11. 11.
    Du S F and Xu M Y, Vertex-primitive \(1/2\)-arc-transitive graphs of smallest order, Commun. Algebra 27 (1999) 163–171MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Feng Y Q and Kwak J H, Cubic symmetric graphs of order twice and odd prime-power, J. Aust. Math. Soc. 81 (2006) 153–164MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Feng Y Q, Kwak J H, Xu M Y and Zhou J X, Tetravalent half-arc-transitive graphs of order \(p^{4}\), Eur. J. Comb. 29 (2008) 555–567CrossRefzbMATHGoogle Scholar
  14. 14.
    Gorenstein D, Finite simple groups, 2nd edition (1982) (New York: Plenum Press) pp. 490–491CrossRefGoogle Scholar
  15. 15.
    Holt D F, A graph which is edge-transitive but not arc transitive, J. Graph Theory 5 (1981) 201–204MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Li C H and Sim H S, On half-transitive metacirculant graphs of prime-power order, J. Comb. Theory B. 81 (2001) 45–57MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Malnič A and Marušič D, Constructing 4-valent 1/2-transitive graphs with a nonsolvable automorphism group, J. Comb. Theory B. 75 (1999) 46–55MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Malnič A and Marušič D, Constructing 1/2-arc-transitivegraphs of valency 4 and vertex stabilizer \(\mathbb{Z}_{2}\times \mathbb{Z}_{2}\), Discrete Math. 245 (2002) 203–216MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Marušič D and Praeger C E, Tetravalent graphs admitting half-transitive group action: Alternating cycles, J. Comb. Theory B. 75 (1999) 188–205MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Tutte W T, Connectivity in graphs (1966) (Toronto: University of Toronto Press)zbMATHGoogle Scholar
  21. 21.
    Taylor D E and Xu M Y, Vertex-primitive 1/2-transitive graphs, J. Aust. Math. Soc. A. 57 (1994) 113–124MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Wang X and Feng Y Q, Hexavalent half-arc-transitive graphs of order \(4p\), Eur. J. Comb. 30 (2009) 1263–1270MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Wang X and Feng Y Q, There exists no tetravalent half-arc-transitive graph of order \(2p^2\), Discrete Math. 310 (2010) 1721–1724MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Wielandt H, Finite permutation groups (1964) (New York: Academic Press)zbMATHGoogle Scholar
  25. 25.
    Xu M Y, Half-transitive graphs of prime-cube order, J. Algebr. Comb. 1 (1992) 275–282MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Xu M Y, Zhang Q H and Zhou J X, Arc-transitive cubic graphs of order \(4p\), Chin. Ann. Math. 25 (2004) 545–554MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Zhou J X and Feng Y Q, Cubic bi-Cayley graphs over abelian groups, Eur. J. Comb. 36 (2014) 679–693MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Zhou J X and Feng Y Q, The automorphisms of bi-Cayley graphs, J. Comb. Theory B. 116 (2016) 504–532MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Indian Academy of Sciences 2018

Authors and Affiliations

  1. 1.Department of MathematicsBeijing Jiaotong UniversityBeijingChina

Personalised recommendations