Extremal number of gluing two copies of a bipartite graph

Determine whether, for two copies H_1 and H_2 of every bipartite graph H and arbitrary vertices u\in V(H_1) and v\in V(H_2), the vertex-glued graph H_1^u\odot H_2^v satisfies ex(n,H_1^u\odot H_2^v)=\Theta(ex(n,H)).

Background

This conjecture is the special case of the general gluing conjecture in which the two constituent graphs are isomorphic copies of a single bipartite graph H. The position of the identified vertices may be in the same or opposite parts of the bipartition.

The paper proves the conjecture when the two identified vertices lie in the same part of H. The unrestricted formulation remains tied, through the paper’s equivalence theorem, to the unresolved comparison between Zarankiewicz and balanced bipartite extremal numbers.

References

The following special case of~\cref{conj: mian} when $H_1, H_2$ are isomorphic is particularly interesting, as we shall see that it is closely related to the famous Zarankiewicz problem.

\begin{conjecture}\label{conj: mian2} If $H_1, H_2$ are two copies of a bipartite graph $H$ and $u \in V(H_1), \, v \in V(H_2)$, then

ex(n, H_1u \odot H_2v) = \Theta\bigl( ex(n, H) \bigr).

— Bipartite Turán problem on graph gluing  (2501.12953 - Dong et al., 22 Jan 2025) in Conjecture 2, Section 1 (Introduction)