Remarks on Path-transitivity in Finite Graphs

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 groupGof a graph Γ is transitive on vertices and on undirected paths of lengthk+1in Γ, for somek≥1, thenGis also transitive onk-arcs in Γ. Further details are given for the casek=1, for the case of cubic graphs, and for the casek>4.