Vertex-gluing extremal-number conjecture

Determine whether, for all bipartite graphs H_1 and H_2 and 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_1)+ex(n,H_2)).

Background

The paper studies the extremal number of a graph formed by identifying one specified vertex of each of two bipartite graphs. The conjecture proposes that this gluing operation preserves, up to constant factors, the larger-order behavior represented by the sum of the two original extremal numbers.

The authors prove that this general gluing conjecture is equivalent to its isomorphic-special-case formulation and to the Erdős–Simonovits conjecture relating the balanced Zarankiewicz number to the balanced bipartite extremal number. Thus, resolving the gluing conjecture would resolve that longstanding Zarankiewicz problem as well.