Let　d　＞=　4　be　an　fixed　integer.　Lee,　Loh　and　Sudakov　(2016)　[17]　conjectured　that　every　directed　graph　D　with　m　arcs　and　minimum　outdegree　at　least　d　admits　a　bipartition　V　(D)　=　V-1　boolean　OR　V-2　such　that　min{e(V-1,　V-2),　e(V-2,　V-1)}　＞=　(　d-1/2(2d　-　1)　+　o(1)　m,　where　e(V-i,　V　(j))　denote　the　number　of　arcs　in　D　from　V-i　to　V　(j)　for　{i,　j}　=　{1,　2}.　Let　Kd,2　denote　the　directed　graph　obtained　by　orienting　each　edge　of　a　bipartite　graph　K2,d　from　the　part　of　size　d　to　the　other　part.　In　this　paper,　we　show　that　the　conjecture　holds　for　&　minus;&　minus;-＞　Kd,2-free　directed　graphs.　Then,　we　prove　that　the　conjecture　holds　under　the　additional　condition　that　the　minimum　indegree　is　also　at　least　d　-1,　which　improves　a　result　given　by　Hou,　Ma,　Yu　and　Zhang.