Partitioning digraphs with outdegree at least 4

TL;DR

This paper investigates how to partition the vertices of directed graphs with minimum outdegree at least 4 to balance the number of arcs between parts, providing new bounds and partial results for the conjectured optimal partition ratio.

Contribution

The paper establishes a new lower bound for the partition ratio in digraphs with outdegree at least 4 and offers partial results for higher outdegree cases, advancing understanding of digraph partitioning.

Findings

Proves that for outdegree 4, the partition ratio c_4 is at least 3/14.

Provides partial results and bounds for cases where outdegree d ≥ 5.

Supports the conjecture on the asymptotic behavior of c_d for large d.

Abstract

Scott asked the question of determining $c_d$ such that if $D$ is a digraph with $m$ arcs and minimum outdegree $d\ge 2$ then $V(D)$ has a partition $V_1, V_2$ such that $\min\left\{e(V_1,V_2),e(V_2, V_1)\right\}\geq c_dm$, where $e(V_1,V_2)$ (respectively, $e(V_2,V_1)$) is the number of arcs from $V_1$ to $V_2$ (respectively, from $V_2$ to $V_1$). Lee, Loh, and Sudakov showed that $c_2=1/6+o(1)$ and $c_3=1/5+o(1)$, and conjectured that $c_d= \frac{d-1}{2(2d-1)}+o(1)$ for $d\ge 4$. In this paper, we show $c_4=3/14+o(1)$ and prove some partial results for $d\ge 5$.