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

Abstract

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 $\hat{I}{»}_m$ such that $d^{++}(v)\ge \hat{I}{»}_m{d}^{+}(v)$, and $\hat{I}{»}_m\to 1$ while $m\to +\infty$. This result generalizes and improves some known results in a sense.