## Abstract

The Directed Grid Theorem, stating that there is a function f such that a directed graph 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, was proved 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 ≥ 1 there exists t = 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.

Original language | English |
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Pages (from-to) | 922-938 |

Number of pages | 17 |

Journal | SIAM Journal on Discrete Mathematics |

Volume | 36 |

Issue number | 2 |

DOIs | |

State | Published - 1 Jan 2022 |

Externally published | Yes |

## Keywords

- bramble
- directed graph
- directed treewidth

## ASJC Scopus subject areas

- General Mathematics

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