Geodesic
4-Transitive Digraphs I: The Structure of Strong 4-Transitive Digraphs
Discussiones Mathematicae. Graph Theory, Tome 33 (2013) no. 2, pp. 247-260

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Let D be a digraph, V (D) and A(D) will denote the sets of vertices and arcs of D, respectively. A digraph D is transitive if for every three distinct vertices u, v,w ∈ V (D), (u, v), (v,w) ∈ A(D) implies that (u,w) ∈ A(D). This concept can be generalized as follows: A digraph is k-transitive if for every u, v ∈ V (D), the existence of a uv-directed path of length k in D implies that (u, v) ∈ A(D). A very useful structural characterization of transitive digraphs has been known for a long time, and recently, 3-transitive digraphs have been characterized. In this work, some general structural results are proved for k-transitive digraphs with arbitrary k ≥ 2. Some of this results are used to characterize the family of 4-transitive digraphs. Also some of the general results remain valid for k-quasi-transitive digraphs considering an additional hypothesis. A conjecture on a structural property of k-transitive digraphs is proposed.
Keywords: digraph, transitive digraph, quasi-transitive digraph, 4-transitive digraph, k-transitive digraph, k-quasi-transitive digraph 
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Hernández-Cruz, César. 4-Transitive Digraphs I: The Structure of Strong 4-Transitive Digraphs. Discussiones Mathematicae. Graph Theory, Tome 33 (2013) no. 2, pp. 247-260. http://geodesic.mathdoc.fr/item/DMGT_2013_33_2_a0/