The surviving rate of digraphs

In this paper, we consider the k-surviving rate of digraphs. Main results are as follows: (1) if G⃗ is a k-degenerate digraph, then ρk(G⃗)≥1k+1; (2) if G⃗ is a planar digraph, then ρ2(G⃗)>140; (3) if G⃗ is a planar digraph without 4-cycles, then ρ1(G⃗)>151.