Comparison with the chromatic clique

Determine whether every fixed graph H of chromatic number c≥2 satisfies Q(Sub_H)=Ω(Q(Clique_c)), together with the corresponding inequality for the search versions of the two problems.

Background

The paper proves the comparison for connected patterns with chromatic number between two and six, and notes that Hadwiger’s conjecture would extend the comparison to all chromatic numbers. The unresolved issue is whether the comparison holds for every fixed pattern of chromatic number at least two without relying on the existence of the corresponding clique minor.

A resolution could require a different reduction from clique detection to general subgraph detection or a direct quantum argument that bypasses clique-minor methods.

References

Does every fixed pattern $H$ of chromatic number $c\ge2$ satisfy

Q(Sub_H)=\Omega\left(Q(Clique_c)\right),

together with the corresponding inequality for search? \Cref{cor:chromatic-comparison} proves both comparisons for connected $H$ with $2\le c\le6$, and Hadwiger's conjecture would extend this to all $c$.

— Superlinear Quantum Query Lower Bounds for Subgraph Detection  (2609.40263 - Gilani et al., 30 Sep 2026) in Section 1, subsection “Open Problems,” paragraph “Comparison with the chromatic clique”