On judicious bipartitions of directed graphs

Abstract

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 thatmin⁡{e(V1,V2),e(V2,V1)}≥(d−12(2d−1)+o(1))m, 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.