On Seymour's and Sullivan's second neighbourhood conjectures

Abstract

AbstractFor a vertex of a digraph, (, respectively) is the number of vertices at distance 1 from (to, respectively) and is the number of vertices at distance 2 from . In 1995, Seymour conjectured that for any oriented graph there exists a vertex such that . In 2006, Sullivan conjectured that there exists a vertex in such that . We give a sufficient condition in terms of the number of transitive triangles for an oriented graph to satisfy Sullivan's conjecture. In particular, this implies that Sullivan's conjecture holds for all orientations of planar graphs and triangle‐free graphs. An oriented graph is an oriented split graph if the vertices of can be partitioned into vertex sets and such that is an independent set and induces a tournament. We also show that the two conjectures hold for some families of oriented split graphs, in particular, when induces a regular or an almost regular tournament.