Skip to main content
SpringerLink
Log in

Two-arc Transitive Near-polygonal Graphs

  • Chapter
Graph Theory in Paris

Part of the book series: Trends in Mathematics ((TM))

  • 1263 Accesses

Abstract

For an integer m ≥ 3, a near m-gonal graph is a pair (Σ,E) consisting of a connected graph Σ and a set E of m-cycles of Σ such that each 2-arc of Σ is contained in exactly one member of E, where a 2-arc of Σ is an ordered triple (σ, τ, ε) of distinct vertices such that τ is adjacent to both σ and ɛ. The graph Σ is called (G, 2)-arc transitive, where G ≤ Aut(Σ), if G is transitive on the vertex set and on the set of 2-arcs of Σ. From a previous study it arises the question of when a (G, 2)-arc transitive graph is a near m-gonal graph with respect to a G-orbit on m-cycles. In this paper we answer this question by providing necessary and sufficient conditions in terms of the stabiliser of a 2-arc.

In memory of Claude Berge

Supported by a Discovery Project Grant (DP0558677) from the Australian Research Council and a Melbourne Early Career Researcher Grant from The University of Melbourne.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Hardcover Book
USD 109.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. N.L. Biggs, Algebraic Graph Theory. 2nd edition, Cambridge University Press, Cambridge, 1993.

    Google Scholar 

  2. J.D. Dixon and B. Mortimer, Permutation Groups. Springer, New York, 1996.

    MATH  Google Scholar 

  3. F. Buekenhout, Diagrams for geometries and groups. J. Combinatorial Theory Ser. A 27 (1979), 121–151.

    Article  MathSciNet  Google Scholar 

  4. A. Gardiner, C.E. Praeger and S. Zhou, Cross ratio graphs. J. London Math. Soc. (2) 64 (2001), no. 2, 257–272.

    MathSciNet  Google Scholar 

  5. M.A. Iranmanesh, C.E. Praeger and S. Zhou, Finite symmetric graphs with two-arc transitive quotients. J. Combinatorial Theory Ser. B 94 (2005), 79–99.

    Article  MathSciNet  Google Scholar 

  6. C.H. Li, C.E. Praeger and S. Zhou, A class of finite symmetric graphs with 2-arc transitive quotients. Math. Proc. Camb. Phil. Soc. 129 (2000), no. 1, 19–34.

    Article  MathSciNet  Google Scholar 

  7. M. Perkel, Bounding the valency of polygonal graphs with odd girth. Canad. J. Math. 31 (1979), 1307–1321.

    MathSciNet  Google Scholar 

  8. M. Perkel, A characterization of J 1 in terms of its geometry. Geom. Dedicata 9 (1980), no. 3, 291–298.

    Article  MathSciNet  Google Scholar 

  9. M. Perkel, A characterization of PSL(2, 31) and its geometry. Canad. J. Math. 32 (1980), no. 1, 155–164.

    MathSciNet  Google Scholar 

  10. M. Perkel, Polygonal graphs of valency four. Congr. Numer. 35 (1982), 387–400.

    MathSciNet  Google Scholar 

  11. M. Perkel, Trivalent polygonal graphs. Congr. Numer. 45 (1984), 45–70.

    MathSciNet  Google Scholar 

  12. M. Perkel, Trivalent polygonal graphs of girth 6 and 7. Congr. Numer. 49 (1985), 129–138.

    MathSciNet  Google Scholar 

  13. M. Perkel, Near-polygonal graphs. Ars Combinatoria 26(A) (1988), 149–170.

    MathSciNet  Google Scholar 

  14. M. Perkel, Some new examples of polygonal and near-polygonal graphs with large girth. Bull. Inst. Combin. Appl. 10 (1994), 23–25.

    MathSciNet  Google Scholar 

  15. M. Perkel, Personal communication, 2005.

    Google Scholar 

  16. M. Perkel, C.E. Praeger, Polygonal graphs: new families and an approach to their analysis. Congr. Numer. 124 (1997), 161–173.

    MathSciNet  Google Scholar 

  17. M. Perkel, C.E. Praeger, R. Weiss, On narrow hexagonal graphs with a 3-homogeneous suborbit. J. Algebraic Combin. 13 (2001), 257–273.

    Article  MathSciNet  Google Scholar 

  18. C.E. Praeger, Finite transitive permutation groups and finite vertex transitive graphs, in: G. Hahn and G. Sabidussi eds., Graph Symmetry (Montreal, 1996, NATO Adv. Sci. Inst. Ser. C, Math. Phys. Sci., 497), Kluwer Academic Publishing, Dordrecht, 1997, pp. 277–318.

    Google Scholar 

  19. S. Zhou, Almost covers of 2-arc transitive graphs. Combinatorica 24(4) (2004), 731–745.

    Article  MathSciNet  Google Scholar 

  20. S. Zhou, Imprimitive symmetric graphs, 3-arc graphs and 1-designs. Discrete Math. 244 (2002), 521–537.

    Article  MathSciNet  Google Scholar 

  21. S. Zhou, Constructing a class of symmetric graphs. European J. Combinatorics 23 (2002), 741–760.

    Article  Google Scholar 

  22. S. Zhou, Symmetric graphs and flag graphs. Monatshefte für Mathematik 139 (2003), 69–81.

    Article  Google Scholar 

  23. S. Zhou, Trivalent 2-arc transitive graphs of type G 12 are near polygonal. reprint, 2006.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics and Statistics, The University of Melbourne, Parkville, VIC, 3010, Australia

    Sanming Zhou

Authors
  1. Sanming Zhou
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Institut Camille Jordan, Université Claude Bernard Lyon 1, 43 boulevard du 11 novembre 1918, 69622, Villeurbanne cedex, France

    Adrian Bondy

  2. Equipe Combinatoire et Optimisation Case 189, Université Pierre et Marie Curie, Paris 6, 75252, Paris Cedex 05, France

    Jean Fonlupt , Jean-Claude Fournier  & Jorge L. Ramírez Alfonsín ,  & 

  3. Laboratoire d’Informatique Fondamentale d’Orléans, Bâtiment IIIA, Rue Léonard de Vinci, B.P. 6759, 45067, Orléans Cedex 2, France

    Jean-Luc Fouquet

Rights and permissions

Reprints and permissions

Copyright information

© 2006 Birkhäuser Verlag Basel/Switzerland

About this chapter

Cite this chapter

Zhou, S. (2006). Two-arc Transitive Near-polygonal Graphs. In: Bondy, A., Fonlupt, J., Fouquet, JL., Fournier, JC., Ramírez Alfonsín, J.L. (eds) Graph Theory in Paris. Trends in Mathematics. Birkhäuser Basel. https://doi.org/10.1007/978-3-7643-7400-6_30

Download citation

Publish with us

Policies and ethics

Search

Navigation