On bipartitions of directed graphs with small semidegree

Abstract

Let D be a directed graph. The minimum semidegree of D is defined to be the minimum value of the minimum outdegree and the minimum indegree of D. For nonempty sets S,T⊆V(D), we use e(S,T) to denote the number of arcs in D from S to T. If D has m arcs and positive minimum semidegree, then we show that D admits a bipartition V(D)=V1∪V2 such that min{e(V1,V2),e(V2,V1)}≥(1∕6+o(1))m. We also prove that if the minimum semidegree is at least two (or three, respectively), then the constant can be increased to 1∕5 (or 3∕14, respectively). These partly answer a question of Hou and Wu (2018).