Constant congestion brambles in directed graphs

TL;DR

This paper demonstrates that relaxing the directed grid to a constant congestion bramble allows for a polynomial bound relating directed treewidth to the existence of large brambles, improving previous bounds.

Contribution

It establishes a polynomial bound on the size of brambles with constant congestion in directed graphs of high directed treewidth.

Findings

Existence of a polynomial bound for brambles with constant congestion

Directed graphs with high directed treewidth contain large low-congestion brambles

Improvement over previous exponential bounds in directed grid theorems

Abstract

The Directed Grid Theorem, stating that there is a function $f$ such that a directed graphs of directed treewidth at least $f(k)$ contains a directed grid of size at least $k$ as a butterfly minor, after being a conjecture for nearly 20 years, has been proven in 2015 by Kawarabayashi and Kreutzer. However, the function $f$ obtained in the proof is very fast growing. In this work, we show that if one relaxes directed grid to bramble of constant congestion, one can obtain a polynomial bound. More precisely, we show that for every $k \geq 1$ there exists $t = \mathcal{O}(k^{48} \log^{13} k)$ such that every directed graph of directed treewidth at least $t$ contains a bramble of congestion at most $8$ and size at least $k$.