恒定拥塞的荆棘
Constant Congestion Brambles
AI总结：
本文研究图论中荆棘的阶与拥塞度，证明树宽至少为k的图存在阶为~Ω(k^{1/2})且拥塞度为2的荆棘，并给出Grohe-Marx下界的紧上界。
摘要：
无向图 $G$ 中的荆棘是一族 $G$ 的连通子图，使得对于荆棘中任意两个子图 $H_1$ 和 $H_2$，要么 $V(H_1) \cap V(H_2) \neq \emptyset$，要么存在 $G$ 的一条边，其一个端点在 $V(H_1)$ 中，另一个端点在 $V(H_2)$ 中。荆棘的阶是能与荆棘中所有元素相交的顶点集的最小大小。\n 荆棘是与树宽对偶的对象：如 Seymour 和 Thomas 所示，无向图 $G$ 中荆棘的最大阶等于 $G$ 的树宽加一。然而，如 Grohe 和 Marx 所示，高阶荆棘可能必然具有指数大小：在常度数 $n$ 顶点扩展图中，对于任意固定的 $\delta\in (0,\frac{1}{2}]$，阶为 $\Omega(n^{1/2+\delta})$ 的荆棘需要大小关于 $\Omega(n^{2\delta})$ 指数增长。另一方面，Grohe 和 Marx 以及 Chekuri 和 Chuzhoy 的结果组合表明，树宽为 $k$ 的图存在阶为 $\widetilde{\Omega}(k^{1/2})$ 且大小为 $\widetilde{\mathcal{O}}(k^{3/2})$ 的荆棘（$\widetilde{\Omega}$ 和 $\widetilde{\mathcal{O}}$ 分别隐藏多对数因子和除数）。\n 在本文中，我们首先通过证明每个树宽至少为 $k$ 的图 $G$ 包含一个阶为 $\widetilde{\Omega}(k^{1/2})$ 且拥塞度为 $2$ 的荆棘来加强第二个界限，即 $G$ 的每个顶点至多包含在荆棘的两个元素中（因此荆棘的大小与其阶成线性关系）。其次，我们为 Grohe 和 Marx 的下界提供紧的上界：对于每个 $\delta\in (0,\frac{1}{2}]$，每个树宽至少为 $k$ 的图 $G$ 包含一个阶为 $\widetilde{\Omega}(k^{1/2+\delta})$ 且大小为 $2^{\widetilde{\mathcal{O}}(k^{2\delta})}$ 的荆棘。
英文摘要：
A bramble in an undirected graph $G$ is a family of connected subgraphs of $G$ such that for every two subgraphs $H_1$ and $H_2$ in the bramble either $V(H_1) \cap V(H_2) \neq \emptyset$ or there is an edge of $G$ with one endpoint in $V(H_1)$ and the second endpoint in $V(H_2)$. The order of the bramble is the minimum size of a vertex set that intersects all elements of a bramble. Brambles are objects dual to treewidth: As shown by Seymour and Thomas, the maximum order of a bramble in an undirected graph $G$ equals one plus the treewidth of $G$. However, as shown by Grohe and Marx, brambles of high order may necessarily be of exponential size: In a constant-degree $n$-vertex expander a bramble of order $Ω(n^{1/2+δ})$ requires size exponential in $Ω(n^{2δ})$ for any fixed $δ\in (0,\frac{1}{2}]$. On the other hand, the combination of results of Grohe and Marx and Chekuri and Chuzhoy shows that a graph of treewidth $k$ admits a bramble of order $\widetildeΩ(k^{1/2})$ and size $\widetilde{\mathcal{O}}(k^{3/2})$. ($\widetildeΩ$ and $\widetilde{\mathcal{O}}$ hide polylogarithmic factors and divisors, respectively.) In this note, we first sharpen the second bound by proving that every graph $G$ of treewidth at least $k$ contains a bramble of order $\widetildeΩ(k^{1/2})$ and congestion $2$, i.e., every vertex of $G$ is contained in at most two elements of the bramble (thus the bramble is of size linear in its order). Second, we provide a tight upper bound for the lower bound of Grohe and Marx: For every $δ\in (0,\frac{1}{2}]$, every graph $G$ of treewidth at least $k$ contains a bramble of order $\widetildeΩ(k^{1/2+δ})$ and size $2^{\widetilde{\mathcal{O}}(k^{2δ})}$.
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