Improved bounds for the graph parameter ell(p,q)

Determine better bounds on the minimum integer ℓ(p,q) such that every graph of sufficiently large treewidth contains pairwise disjoint connected subgraphs of treewidth at least p whose simultaneous contraction yields a minor of treewidth at least q, with particular attention to the proposed bound ℓ(p,q) ∈ O*(pq).

Background

The paper defines ℓ(p,q) as the minimum threshold guaranteeing both a collection of n pairwise disjoint connected subgraphs, each of treewidth at least p, and a minor of treewidth at least q after contracting those subgraphs. A grid-minor argument gives the bound ℓ(p,q) ∈ O*((pq)9), while the authors propose the conjectural improvement ℓ(p,q) ∈ O*(pq). They explicitly ask whether better bounds are possible; the proposed conjecture would also strengthen a conjecture of Chuzhoy and Chekuri concerning vertex-disjoint subgraphs of large treewidth.

References

Our second open problem is inspired by the proof of \cref{key}, which implicitly relies on the following fact: For any integers $p,q 1$ there is a minimum integer $\ell(p,q)$ such that for any graph $G$ with treewidth at least $\ell(p,q)$: \begin{enumerate}[(a)] \item there are pairwise disjoint connected subgraphs $G_1,G_2,\dots,G_n$ such that $\tw(G_i) p$ for each $i\in{1,\dots,n}$, and \item contracting each of $G_1,\dots,G_n$ to a vertex gives a minor of $G$ with treewidth at least $q$. \end{enumerate} We claim that $\ell(p,q) g( pq)$. Say $\tw(G) g(pq)$. So the $pq\times pq$ grid is a minor of $G$. Partition this grid into $q2$ copies of the $p\times p$ grid. Each subgrid has treewidth $p$ and contracting each subgrid gives a $q\times q$ grid, which has treewidth $q$. By GMT, $\ell(p,q) O\ast( (pq)9 )$. Are there better bounds on $\ell$?

— Polynomial Bounds in the Apex Minor Theorem  (2503.04228 - Hendrey et al., 6 Mar 2025) in Section Open Problems