3 research outputs found

    Sharply kk-arc-transitive-digraphs: finite and infinite examples

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    A general method for constructing sharply kk-arc-transitive digraphs, i.e. digraphs that are kk-arc-transitive but not (k+1)(k+1)-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

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    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

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    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
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