Bisections of directed graphs without complete bipartite subgraphs

It is well-known that every digraph (directed graph) D has a directed cut of size at least $e\left(D\right)/4$, and the constant 1/4 cannot be replaced by any larger one. In this paper, motivated by a problem of Scott (2005) [26] and a conjecture of Lee, Loh and Sudakov (2016) [18], we study bisections of digraphs, concentrating on the situation where a large number of arcs cross the bisection in each direction. For any integers $d\ge s\ge 2$, let $\overrightarrow{K_{d,s}}$ denote the digraph obtained by orienting each edge of the bipartite graph $K_{d,s}$ from the part of size d to the other part. Let D be a digraph with m arcs and minimum outdegree at least d. We prove that if D does not contain $\overrightarrow{K_{d,s}}$, then D admits a bisection in which at least $(\frac{2d^2-2ds+2d-s+2}{4d(2d-2s+3)}+o(1\left)\right)m$ arcs cross the bisection in each direction. Moreover, if the underlying graph of D does not contain triangles, we show that D admits a bisection in which at least $(1/4+o(1\left)\right)m$ arcs cross the bisection in each direction.