The Second Neighbourhood for Quasi-transitive Oriented Graphs

In 2006, Sullivan stated the conjectures: (1) every oriented graph has a vertex x such that d ++ ( x ) ≥ d − ( x ); (2) every oriented graph has a vertex x such that d ++ ( x )+ d + ( x ) ≥ 2 d − ( x ); (3) every oriented graph has a vertex x such that d ++ ( x ) + d + ( x ) ≥ 2 · min{ d + ( x ), d − ( x )}. A vertex x in D satisfying Conjecture ( i ) is called a Sullivan- i vertex, i = 1, 2, 3. A digraph D is called quasi-transitive if for every pair xy , yz of arcs between distinct vertices x , y , z , xz or zx (“or” is inclusive here) is in D . In this paper, we prove that the conjectures hold for quasi-transitive oriented graphs, which is a superclass of tournaments and transitive acyclic digraphs. Furthermore, we show that a quasi-transitive oriented graph with no vertex of in-degree zero has at least three Sullivan-1 vertices and a quasi-transitive oriented graph has at least three Sullivan-3 vertices unless it belongs to an exceptional class of quasi-transitive oriented graphs. For Sullivan-2 vertices, we show that an extended tournament, a subclass of quasi-transitive oriented graphs and a superclass of tournaments, has at least two Sullivan-2 vertices unless it belongs to an exceptional class of extended tournaments.