Optimally large cuts in both directions

Abstract

We prove that for every integer $d\ge 2$ and all sufficiently large $m$, every digraph with $m$ edges and minimum outdegree at least $d$ has a vertex bipartition for which each directed cut is of size at least \[\left\lceil\frac{d-1}{2(2d-1)}m+\frac{d^2}{2(2d-1)}\right\rceil.\] The bound is sharp for infinitely many $m$. This answers for large graphs a question of Scott and proves a conjecture of Lee, Loh, and Sudakov. For $d=2$ we prove that $\lceil m/6\rceil$ is achievable for all $m$. We also establish a stability result characterizing near-extremal graphs.