Bipartite Turán problem on graph gluing

Abstract: For graphs $H_1$ and $H_2$, if we glue them by identifying a given pair of vertices $u \in V(H_1)$ and $v \in V(H_2)$, what is the extremal number of the resulting graph $H_1^u \odot H_2^v$? In this paper, we study this problem and show that interestingly it is equivalent to an old question of Erd\H{o}s and Simonovits on the Zarankiewicz problem. When $H_1, H_2$ are copies of a same bipartite graph $H$ and $u, v$ come from a same part, we prove that $\operatorname{ex}(n, H_1^u \odot H_2^v) = \Theta \bigl( \operatorname{ex}(n, H) \bigr)$. As a corollary, we provide a short self-contained disproof of a conjecture of Erd\H{o}s, which was recently disproved by Janzer.