Bipartite Tur\'an problems via graph gluing

TL;DR

This paper investigates the extremal number of graphs formed by gluing bipartite graphs at vertices, linking it to the Zarankiewicz problem, and proves a key asymptotic equivalence with implications for longstanding conjectures.

Contribution

It establishes a connection between bipartite graph gluing and the Zarankiewicz problem, providing new asymptotic results and a simplified proof of a conjecture of Erdős.

Findings

Proves the extremal number for glued bipartite graphs is asymptotically equivalent to that of the original graph.

Links the problem to the classical Zarankiewicz problem, revealing new insights.

Provides a simplified disproof of a longstanding Erdős conjecture.

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.