Linear-order conjecture for ell(p,q)

Prove or refute the conjecture that the minimum integer ℓ(p,q) guaranteeing pairwise disjoint connected subgraphs of treewidth at least p whose contractions produce a minor of treewidth at least q satisfies ℓ(p,q) ∈ O*(pq).

Background

After deriving the O*((pq)9) bound, the authors propose the substantially stronger conjecture ℓ(p,q) ∈ O*(pq). They note that this conjecture would imply and strengthen an earlier conjecture concerning the number of vertex-disjoint subgraphs of treewidth p. The conjecture is explicitly presented as part of the discussion of the second open problem.

References

Indeed, \cref{con1} would imply and strengthen Conjecture 1 in , which considers only the number of vertex-disjoint graphs of treewidth $p$.

— Polynomial Bounds in the Apex Minor Theorem  (2503.04228 - Hendrey et al., 6 Mar 2025) in Conjecture 1, Section Open Problems