Skip to main content
SpringerLink
Log in

Highly arc-transitive digraphs — Structure and counterexamples

  • Original Paper
  • Published:
Combinatorica Aims and scope Submit manuscript

Abstract

Two problems of Cameron, Praeger, and Wormald [Infinite highly arc transitive digraphs and universal covering digraphs, Combinatorica (1993)] are resolved. First, locally finite highly arc-transitive digraphs with universal reachability relation are presented. Second, constructions of two-ended highly arc-transitive digraphs are provided, where each ‘building block’ is a finite bipartite digraph that is not a disjoint union of complete bipartite digraphs. Both of these were conjectured impossible in the above-mentioned paper. We also describe the structure of two-ended highly arc-transitive digraphs in more generality, heading towards a characterization of such digraphs. However, the complete characterization remains elusive.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. L. Bartholdi, M. Neuhauser and W. Woess: Horocyclic products of trees, J. Eur. Math. Soc. (JEMS) 10 (2008), 771–816.

    Article  MATH  MathSciNet  Google Scholar 

  2. J. A. Bondy and U. S. R. Murty: Graph theory, Graduate Texts in Mathematics, vol. 244, Springer, New York, 2008.

  3. P. J. Cameron, C. E. Praeger and N. C. Wormald: Infinite highly arc transitive digraphs and universal covering digraphs, Combinatorica 13 (1993), 377–396.

    Article  MATH  MathSciNet  Google Scholar 

  4. R. Diestel: Graph theory, third ed., Graduate Texts in Mathematics, vol. 173, Springer-Verlag, Berlin, 2005.

  5. M. J. Dunwoody: Cutting up graphs, Combinatorica 2 (1982), 15–23.

    Article  MATH  MathSciNet  Google Scholar 

  6. A. Malnič, D. Marušič, R. G. Möller, N. Seifter, V. Trofimov and B. Zgrablič: Highly arc transitive digraphs: reachability, topological groups, European Journal of Combinatorics 26 (2005), 19–28.

    Article  MATH  MathSciNet  Google Scholar 

  7. A. Malnič, D. Marušič, N. Seifter and B. Zgrablić: Highly arc-transitive di-graphs with no homomorphism onto ℤ, Combinatorica 22 (2002), 435–443.

    Article  MATH  MathSciNet  Google Scholar 

  8. R. G. Möller: Descendants in highly arc transitive digraphs, Discrete Mathematics 247 (2002), 147–157.

    Article  MATH  MathSciNet  Google Scholar 

  9. R. G. Möller: Structure theory of totally disconnected locally compact groups via graphs and permutations, Canad. J. Math. 54 (2002), 795–827.

    Article  MATH  MathSciNet  Google Scholar 

  10. C. Neumann: Constructing highly arc transitive digraphs using a direct fibre product, Discr. Math. 313 (2013), 2816–2829.

    Article  MATH  Google Scholar 

  11. C. E. Praeger: On homomorphic images of edge transitive directed graphs, Australas. J. Combin. 3 (1991), 207–210.

    MATH  MathSciNet  Google Scholar 

  12. G. Sabidussi: Vertex-transitive graphs, Monatsh. Math. 68 (1964), 426–438.

    Article  MATH  MathSciNet  Google Scholar 

  13. W. T. Tutte: A family of cubical graphs, Proc. Cambridge Philos. Soc. 43 (1947), 459–474.

    Article  MATH  MathSciNet  Google Scholar 

  14. R. Weiss: The nonexistence of 8-transitive graphs, Combinatorica 1 (1981), 309–311.

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Dept. of Mathematics, Simon Fraser University, Burnaby, Canada

    Matt DeVos & Bojan Mohar

  2. Computer Science Institute of Charles University, Prague, Czech Republic

    Robert Šámal

Authors
  1. Matt DeVos
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Bojan Mohar
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Robert Šámal
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Robert Šámal.

Additional information

Supported in part by the Research Grant P1-0297 of ARRS (Slovenia), by an NSERC Discovery Grant (Canada) and by the Canada Research Chair program.

On leave from: IMFM & FMF, Department of Mathematics, University of Ljubljana, Ljubljana, Slovenia.

Partially supported by Karel Janeček Science & Research Endowment (NFKJ) grant 201201. Partially supported by grant LL1201 ERC CZ of the Czech Ministry of Education, Youth and Sports. Partially supported by grant GA ČR P202-12-G061.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

DeVos, M., Mohar, B. & Šámal, R. Highly arc-transitive digraphs — Structure and counterexamples. Combinatorica 35, 553–571 (2015). https://doi.org/10.1007/s00493-014-3040-4

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00493-014-3040-4

Mathematics Subject Classification (2010)

Search

Navigation