3 research outputs found
Sharply -arc-transitive-digraphs: finite and infinite examples
A general method for constructing sharply -arc-transitive digraphs, i.e.
digraphs that are -arc-transitive but not -arc-transitive, is
presented. Using our method it is possible to construct both finite and
infinite examples. The infinite examples can have one, two or infinitely many
ends. Among the one-ended examples there are also digraphs that have polynomial
growth
Infinite arc-transitive and highly-arc-transitive digraphs
A detailed description of the structure of two-ended arc-transitive digraphs
is given. It is also shown that several sets of conditions, involving such
concepts as Property Z, local quasi-primitivity and prime out-valency, imply
that an arc-transitive digraph must be highly-arc-transitive. These are then
applied to give a complete classification of two-ended highly-arc-transitive
digraphs with prime in- and out-valencies.Comment: To appear in European Journal of Combinatorics. Statements of
Corollaries 14, 16 and 17 correcte
Highly arc-transitive digraphs -- counterexamples and structure
We resolve two problems of [Cameron, Praeger, and Wormald -- Infinite highly
arc transitive digraphs and universal covering digraphs, Combinatorica 1993].
First, we construct a locally finite highly arc-transitive digraph with
universal reachability relation. Second, we provide constructions of 2-ended
highly arc transitive digraphs where each `building block' is a finite
bipartite graph that is not a disjoint union of complete bipartite graphs. This
was conjectured impossible in the above paper. We also describe the structure
of 2-ended highly arc transitive digraphs in more generality, although complete
characterization remains elusive.Comment: 18 page