Highly-Arc-Transitive and Descendant-Homogeneous Digraphs with Finite Out-Valency


We investigate infinite highly-arc-transitive digraphs with two additional properties, property Z and descendant-homogeneity. We show that if D is a highly-arc-transitive descendant-homogeneous digraph with property Z and F is the subdigraph spanned by the descendant sets of a line in D, then F is a locally finite 2-ended digraph with property Z. If, moreover, D has prime out-valency, then there is only one possibility for the subdigraph F. This knowledge is then used to classify the highly-arc-transitive descendant-homogeneous digraphs of prime out-valency with property Z.

This is a preview of subscription content, access via your institution.


  1. [1]

    D. Amato: Descendants in infinite, primitive, highly-arc-transitive digraphs, Discrete Mathematics310 (2010), 2021–2036.

    MATH  Google Scholar 

  2. [2]

    D. Amato and J. K. Truss: Descendant-homogeneous digraphs, Journal of Combinatorial Theory, Series A31 (2011), 247–283.

    MathSciNet  MATH  Google Scholar 

  3. [3]

    D. Amato and D. M. Evans: Infinite primitive and distance transitive directed graphs of finite out-valency, Journal of Combinatorial Theory, Series B114 (2015), 33–50.

    MathSciNet  Article  Google Scholar 

  4. [4]

    M. DeVos, B. Mohar and R. Sámal: Highly arc-transitive digraphs - structure and counterexamples, Combinatorica35 (2015), 553–571.

    MathSciNet  Article  Google Scholar 

  5. [5]

    P. J. Cameron, Ch. E. Praeger and N. C. Wormald: Infinite highly arc transitive digraphs and universal covering digraphs, Combinatorica13 (1993), 377–396.

    MathSciNet  Article  Google Scholar 

  6. [6]

    C. W. H. Lam: Distance transitive digraphs, Discrete Mathematics29 (1980), 265–274.

    MathSciNet  Article  Google Scholar 

  7. [7]

    R. G. Möller: Descendants in highly arc transitive digraphs, Discrete Mathematics247 (2002), 147–157.

    MathSciNet  Article  Google Scholar 

  8. [8]

    R. G. Möller, P. Potočnik and N. Seifter: Infinite arc-transitive and highly-arc-transitive digraphs, European Journal of Combinatorics77 (2019), 78–89.

    MathSciNet  Article  Google Scholar 

Download references

Author information



Corresponding author

Correspondence to Daniela A. Amato.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Amato, D.A. Highly-Arc-Transitive and Descendant-Homogeneous Digraphs with Finite Out-Valency. Combinatorica 39, 1203–1223 (2019). https://doi.org/10.1007/s00493-019-3906-6

Download citation

  • 05C20
  • 05E18
  • 20B07
  • 20B27