Abstract
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.
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References
D. Amato: Descendants in infinite, primitive, highly-arc-transitive digraphs, Discrete Mathematics310 (2010), 2021–2036.
D. Amato and J. K. Truss: Descendant-homogeneous digraphs, Journal of Combinatorial Theory, Series A31 (2011), 247–283.
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.
M. DeVos, B. Mohar and R. Sámal: Highly arc-transitive digraphs - structure and counterexamples, Combinatorica35 (2015), 553–571.
P. J. Cameron, Ch. E. Praeger and N. C. Wormald: Infinite highly arc transitive digraphs and universal covering digraphs, Combinatorica13 (1993), 377–396.
C. W. H. Lam: Distance transitive digraphs, Discrete Mathematics29 (1980), 265–274.
R. G. Möller: Descendants in highly arc transitive digraphs, Discrete Mathematics247 (2002), 147–157.
R. G. Möller, P. Potočnik and N. Seifter: Infinite arc-transitive and highly-arc-transitive digraphs, European Journal of Combinatorics77 (2019), 78–89.