排除网格定理的改进界

摘要：

本文研究了Robertson和Seymour的排他网格定理。这是图论中的一个基本结果，即存在某个函数$f: Z^+\rightarrow Z^+$，使得对于所有整数$g>0$，每个树宽至少为$f(g)$的图都包含$(g\times g)$ -网格作为子图。直到最近，在$g$中，关于$f$的最佳已知上界是超指数的。Chekuri和Chuzhoy最近的一项工作提供了第一个多项式界，通过证明树宽$f(g)=O(g^{98}\operatorname{poly}\log g)$足以确保任何图中存在$(g\times g)$ -网格子式。在本文中，我们改进了这个界限$f(g)=O(g^{19}\operatorname{poly}\log g)$。我们引入了一些新的技术，包括一个概念上简单，几乎完全自包含的定理证明，实现了多项式约束$f(g)$。

英文摘要：

We study the Excluded Grid Theorem of Robertson and Seymour. This is a fundamental result in graph theory, that states that there is some function $f: Z^+\rightarrow Z^+$, such that for all integers $g>0$, every graph of treewidth at least $f(g)$ contains the $(g\times g)$-grid as a minor. Until recently, the best known upper bounds on $f$ were super-exponential in $g$. A recent work of Chekuri and Chuzhoy provided the first polynomial bound, by showing that treewidth $f(g)=O(g^{98}\operatorname{poly}\log g)$ is sufficient to ensure the existence of the $(g\times g)$-grid minor in any graph. In this paper we improve this bound to $f(g)=O(g^{19}\operatorname{poly}\log g)$. We introduce a number of new techniques, including a conceptually simple and almost entirely self-contained proof of the theorem that achieves a polynomial bound on $f(g)$.

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