Juan Liu, Hong Yang, Xindong Zhang, Hong-Jian Lai
For a vertex $x$ of a digraph $D$, $|N_{D}^{+}(x)|$ is the number of vertices at distance 1 from $x$ and $|N_{D}^{++}(x)|$ is the number of vertices at distance 2 from $x$. In 1990, Seymour conjectured that for any oriented graph $D$ there exists a vertex $x$ such that $|N_{D}^{+}(x)| \leq |N_{D}^{++}(x)|$, where $x$ is called Seymour vertex. In 2018, Dara et al. conjectured that in every oriented graph with no sink, there are at least two Seymour vertices. In this paper, we investigate the existence of a Seymour vertex in line digraph and give a sufficient and necessary condition for line digraph to have a Seymour vertex. In particular, the result that line digraph of oriented graph has a Seymour vertex is obtained. Moreover, we give a sufficient and necessary condition for jump digraph (complement of line digraph) of digraph to have a Seymour vertex or at least two Seymour vertices, respectively.
