有向图中的常拥堵网状结构
Constant congestion brambles in directed graphs
AI总结：
本研究证明，通过将有向网格改为常拥堵的bramble，有向图的树宽与bramble大小之间的关系可获得多项式界。
摘要：
有向网格定理指出，存在一个函数f，使得有向树宽至少为f(k)的有向图包含一个大小至少为k的有向网格作为蝴蝶 minors。该定理在近20年后于2015年由Kawarabayashi和Kreutzer证明。然而，证明中得到的函数f增长非常迅速。在本工作中，我们证明如果将有向网格放松为常拥堵的bramble，则可以得到多项式界。更具体地说，对于每一个k≥1，存在t=O(k^{48} log^{13}k)，使得每个有向树宽至少为t的有向图包含一个拥堵至多为8且大小至少为k的bramble。
英文摘要：
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 $k$.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
