Meike Hatzel, Constant congestion bramble

In this talk I will present a small result we achieved during a workshop in February this year. My coauthors on this are Marcin Pilipczuk, Paweł Komosa and Manuel Sorge.

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 $V(H_1)\cap V(H_2)\ne \mathrm{\varnothing }$ 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 $\mathrm{\Omega }(n^{1/2+\delta })$ requires size exponential in $\mathrm{\Omega }(n^{2\delta })$ for any fixed $\delta \in (0,\frac{1}{2}]$. 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 $\tilde{\mathrm{\Omega }}(k^{1/2})$ and size $\tilde{O}(k^{3/2})$. ($\tilde{\mathrm{\Omega }}$ and $\tilde{O}$ hide polylogarithmic factors and divisors, respectively.)

We first sharpen the second bound by proving that every graph $G$ of treewidth at least $k$ contains a bramble of order $\tilde{\mathrm{\Omega }}(k^{1/2})$ 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 $\delta \in (0,\frac{1}{2}]$, every graph $G$ of treewidth at least $k$ contains a bramble of order $\tilde{\mathrm{\Omega }}(k^{1/2+\delta })$ and size $2^{\tilde{O}(k^{2\delta })}$.