Optimal bisections of directed graphs

TL;DR

This paper investigates bisections in directed graphs, establishing an optimal bound on the number of crossing arcs based on minimum semidegree, thus advancing understanding of graph partitioning.

Contribution

It proves an optimal bound for directed graph bisections related to minimum semidegree, answering a question by Hou and Wu with a stronger result.

Findings

Every directed graph with m arcs and minimum semidegree d has a bisection with at least (d/2(2d+1)+o(1))m crossing arcs in each direction.

The result confirms the bound's optimality.

Provides a positive answer to a conjecture by Lee, Loh, and Sudakov.

Abstract

In this paper, motivated by a problem of Scott and a conjecture of Lee, Loh and Sudakov we consider bisections of directed graphs. We prove that every directed graph with $m$ arcs and minimum semidegree at least $d$ admits a bisection in which at least $\left(\frac{d}{2(2d+1)}+o(1)\right)m$ arcs cross in each direction. This provides an optimal bound as well as a positive answer to a question of Hou and Wu in a stronger form.