@inbook{16aa64efb319455db13d9288263fc8c7,

title = "Constant Congestion Brambles in Directed Graphs",

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, has been proven in 2015 by Kawarabayashi and Kreutzer. However, the function f in the proof is very fast growing. Here, we show that if we relax directed grid to bramble of constant congestion, we obtain a polynomial bound. More precisely, we show that for every k≥ 1 there exists t= O(k48log13k) such that every directed graph of directed treewidth at least t contains a bramble of congestion at most 8 and size at least k.",

keywords = "Directed graphs, Directed treewidth, Graph theory",

author = "Tom{\'a}{\v s} Masa{\v r}{\'i}k and Marcin Pilipczuk and Pawe{\l} Rz{\c a}{\.z}ewski and Manuel Sorge",

note = "Funding Information: This research is part of projects that have received funding from the European Research Council (ERC) under the European Union{\textquoteright}s Horizon 2020 research and innovation programme Grant Agreement 714704. T.M. was partially supported by a postdoctoral fellowship at the Simon Fraser University through NSERC grants R611450 and R611368. M.S. was partially supported by Alexander von Humboldt Foundation. The full version of the paper can be found on arXiv [11]. Publisher Copyright: {\textcopyright} 2021, The Author(s), under exclusive license to Springer Nature Switzerland AG.",

year = "2021",

month = jan,

day = "1",

doi = "10.1007/978-3-030-83823-2_50",

language = "English",

series = "Trends in Mathematics",

publisher = "Springer Science and Business Media Deutschland GmbH",

pages = "318--324",

booktitle = "Trends in Mathematics",

address = "Germany",

}