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A fascinating problem on digraphs is the existence problem of the finiteupperbound on s for all vertex-primitive s-arc-transitive digraphs except directedcycles (which is known to be reduced to the almost simple groups case). In thispaper,we prove that s 2 for all G-vertex-primitive s-arc-transitive digraphswith G an (insoluble) alternating or symmetric group, which makes an importantprogress towards a solution of the problem. The proofs involves some methodsthat may be used to investigate other almost simple groups cases.

Vertex-primitive s-arc-transitive digraphs of alternating and symmetric groups