Bipartitions of oriented graphs

Let V(D)=X∪Y be a bipartition of a directed graph D. We use e(X,Y) to denote the number of arcs in D from X to Y. Motivated by a conjecture posed by Lee, Loh and Sudakov (2016) [16], we study bipartitions of oriented graphs. Let D be an oriented graph with m arcs. In this paper, it is proved that if the minimum degree of D is δ, then D admits a bipartition V(D)=V1∪V2 such that min⁡{e(V1,V2),e(V2,V1)}≥(δ−14δ+o(1))m. Moreover, if the minimum semidegree d=min⁡{δ+(D),δ−(D)} of D is at least 21, then D admits a bipartition V(D)=V1∪V2 such that min⁡{e(V1,V2),e(V2,V1)}≥(d2(2d+1)+o(1))m. Both bounds are asymptotically best possible.