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)=V1∪V2　such　that　[Formula　presented]　where　e(Vi,Vj)　denote　the　number　of　arcs　in　D　from　Vi　to　Vj　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　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.　©　2022　Elsevier　B.V.