Subgraphs with a positive minimum semidegree in digraphs with large outdegree

Abstract

We prove that every $n$-vertex directed graph $G$ with the minimum outdegree $\hat{I}{´}^{+}(G)=d$ contains a subgraph $H$ satisfying \[ \min\left\{δ^+(H), δ^-(H) \right\} \ge \frac{d(d+1)}{2n} \,.\] We also show that if $d=o(n)$ then this bound is asymptotically best possible.