Abstract
This paper deals with graphs the automorphism groups of which are transitive on vertices and on undirected paths (but not necessarily on directed walks) of some fixed length. In particular, it is shown that if the automorphism group Gof a graph Γ is transitive on vertices and on undirected paths of length k+1in Γ, for some k≥1, then Gis also transitive on k-arcs in Γ. Further details are given for the case k=1, for the case of cubic graphs, and for the case k>4.
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