Profile and neighbourhood complexity for excluded complete minors

Determine, up to a constant factor, the profile and neighbourhood complexity of graphs excluding a complete graph K_t as a minor.

Background

The paper improves the known upper bound for the r-profile complexity of graphs excluding K_h as a minor to h{O(h)}(r+1){3(h-1)}k. Despite this improvement, the authors state that the optimal dependence, up to a constant factor, of both profile and neighbourhood complexity for K_t-minor-free graphs remains unresolved.

References

While Theorem~\ref{thm:profilesminors} improves on the upper bound for the profile (and neighbourhood) complexity of graphs excluding a fixed complete minor, the following remains open. Up to a constant factor, what are the profile and neighbourhood complexity of graphs excluding $K_t$ as a minor?

— Profile and neighbourhood complexity of graphs with excluded minors and tree-structured graphs  (2501.08895 - Beaudou et al., 15 Jan 2025) in Section 6, “Open problems,” subsection “Graphs excluding a fixed minor,” immediately after Theorem profilesminors