On Seymour's Second Neighborhood Conjecture of m-free Digraphs

This paper gives an approximate result related to Seymour's Second Neighborhood conjecture, that is, for any $m$-free digraph $G$, there exists a vertex $v\in V(G)$ and a real number $\lambda_m$ such that $d^{++}(v)\geq \lambda_m d^+(v)$, and $\lambda_m \rightarrow 1$ while $m \rightarrow +\infty$. This result generalizes and improves some known results in a sense.