Small complete bipartite subgraph detection

Determine superlinear quantum query lower bounds for detecting complete bipartite graphs K_{r,r} with r<10, including the four-cycle K_{2,2}.

Background

The paper proves superlinear lower bounds for detecting K_{r,r} only once r reaches 10, with the exponent becoming superlinear at that threshold. Consequently, the methods do not resolve the complexity of smaller complete bipartite patterns.

The four-cycle K_{2,2} is highlighted as a specific unresolved case, making it a basic target for extending superlinear quantum query lower bounds to small bipartite subgraphs.

References

Our bounds for complete bipartite graphs also leave small patterns open. The exponent for $K_{r,r}$ becomes superlinear at $r=10$. Smaller patterns, including the four-cycle $K_{2,2}$, remain open.

— Superlinear Quantum Query Lower Bounds for Subgraph Detection  (2609.40263 - Gilani et al., 30 Sep 2026) in Section 1, subsection “Open Problems,” paragraph “Triangles and small bipartite patterns”