Generalized Tur\'an problems for complete bipartite graphs

TL;DR

This paper investigates the maximum number of copies of a complete bipartite graph within an $F$-free graph, focusing on the generalized Turán number for such bipartite graphs.

Contribution

It provides new bounds and insights into the generalized Turán problem specifically for complete bipartite graphs, extending classical extremal graph theory results.

Findings

Derived bounds for $ex(n,G,F)$ when $G$ and $F$ are complete bipartite graphs

Identified extremal constructions for maximizing copies of $G$

Extended classical Turán results to a generalized bipartite setting

Abstract

For graph $G$, $F$ and integer $n$, the generalized Tu\'an number $ex(n,G,F)$ denotes the maximum number of copies of $G$ that an $F$-free $n$-vertex graph can have. We study this parameter when both $G$ and $F$ are complete bipartite graphs.