  For a vertex $x$ of a digraph, $d^+(x)$ ($d^-(x)$, resp.) is the number of
vertices at distance 1 from (to, resp.) $x$ and $d^{++}(x)$ is the number of
vertices at distance 2 from $x$. In 1995, Seymour conjectured that for any
oriented graph $D$ there exists a vertex $x$ such that $d^+(x)\leq d^{++}(x)$.
In 2006, Sullivan conjectured that there exists a vertex $x$ in $D$ such that
$d^-(x)\leq d^{++}(x)$. 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 of triangle-free graphs. An oriented graph
$D$ is an oriented split graph if the vertices of $D$ can be partitioned into
vertex sets $X$ and $Y$ such that $X$ is an independent set and $Y$ induces a
tournament. We also show that the two conjectures hold for some families of
oriented split graphs, in particular, when $Y$ induces a regular or an almost
regular tournament.
