Highly arc-transitive digraphs — Structure and counterexamples

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.

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Correspondence to Robert Šámal.

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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.

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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

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Mathematics Subject Classification (2010)

  • 05E99
  • 05C25
  • 05C20
  • 05C63