Partitioning digraphs with outdegree at least 4

Scott asked the question of determining $c_d$ such that if $D$ is a digraph with $m$ arcs and minimum outdegree $d\ge 2$ then $V(D)$ has a partition $V_1, V_2$suchthat$\min\left{e(V_1,V_2),e(V_2, V_1)\right}\geq c_dm$,where$e(V_1,V_2)$(respectively,$e(V_2,V_1)$)isthenumberofarcsfrom$V_1$to$V_2$(respectively,from$V_2$to$V_1$)\mathrm{.}Lee,Loh,andSudakovshowedthat$c_2=1/6+o(1)$and$c_3=1/5+o(1)$,andconjecturedthat$c_d= \frac{d-1}{2(2d-1)}+o(1)$for$d\ge 4$\mathrm{.}Inthispaper,weshow$c_4=3/14+o(1)$andprovesomepartialresultsfor$d\ge 5$.