  Judicious partitioning problems on graphs ask for partitions that bound
several quantities simultaneously, which have received a lot of attentions
lately. Scott asked the following natural question: What is the maximum
constant $c_d$ such that every directed graph $D$ with $m$ arcs and minimum
outdegree $d$ admits a bipartition $V(D)= V_1\cup V_2$ satisfying $\min\{e(V_1,
V_2), e(V_2, V_1)\}\ge c_d m$? Here, for $i=1,2$, $e(V_{i},V_{3-i})$ denotes
the number of arcs in $D$ from $V_{i}$ to $V_{3-i}$. Lee, Loh, and Sudakov
%[Judicious partitions of directed graphs, Random Struct. Alg. 48 %(2016)
147--170] conjectured that every directed graph $D$ with $m$ arcs and minimum
outdegree at least $d\ge 2$ admits a bipartition $V(D)=V_1\cup V_2$ such that
\[ \min\{e(V_1,V_2),e(V_2,V_1)\}\geq \Big(\frac{d-1}{2(2d-1)}+ o(1)\Big)m. \]
%While it is not known whether or not the minimum outdegree condition %alone is
sufficient, w We show that this conjecture holds under the additional natural
condition that the minimum indegree is also at least $d$.
