Generalized Tur\'an problems for disjoint copies of graphs

TL;DR

This paper studies the maximum number of copies of a graph H in large graphs that avoid k disjoint copies of another graph F, extending Turán-type extremal problems to multiple forbidden subgraphs.

Contribution

It introduces bounds and exact results for ex(n,H,kF) for various classes of graphs F, generalizing classical Turán problems to multiple disjoint forbidden subgraphs.

Findings

Derived bounds for ex(n,H,kF) when F is complete, cycle, or bipartite.

Provided exact extremal numbers for specific graph classes.

Extended Turán theory to multiple disjoint forbidden subgraphs.

Abstract

Given two graphs $H$ and $F$, the maximum possible number of copies of $H$ in an $F$-free graph on $n$ vertices is denoted by $ex(n,H,F)$. We investigate the function $ex(n,H,kF)$, where $kF$ denotes $k$ vertex disjoint copies of a fixed graph $F$. Our results include cases when $F$ is a complete graph, cycle or a complete bipartite graph.