Constant Congestion Brambles
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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) ∩ V(H_2) ≠∅ 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∈ (0,1/2]. 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 Ω(k^1/2) and size 𝒪(k^3/2). (Ω and 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 Ω(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∈ (0,1/2], every graph G of treewidth at least k contains a bramble of order Ω(k^1/2+δ) and size 2^𝒪(k^2δ).
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