Discussiones Mathematicae Graph Theory 33(2) (2013) 247-260
doi: 10.7151/dmgt.1645

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4-transitive Digraphs I: The Structure of Strong 4-transitive Digraphs

César Hernández-Cruz

Instituto de Matemáticas
Universidad Nacional Autónoma de México
Ciudad Universitaria, México, D.F., C.P. 04510, México

Abstract

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

2010 Mathematics Subject Classification: 05C20, 05C75.

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Received 25 May 2011
Revised 22 February 2012
Accepted 23 February 2012