Generalized Tur\'an results for matchings

TL;DR

This paper investigates the maximum number of copies of a graph H in large graphs that avoid another graph F, focusing on cases where either H or F is a matching, providing new asymptotic and exact results.

Contribution

It introduces new asymptotic and exact results for the generalized Turán number when either H or F is a matching, expanding understanding of extremal graph configurations.

Findings

Derived asymptotic formulas for specific matchings and graphs.

Established exact extremal numbers in certain cases.

Extended classical Turán results to generalized settings.

Abstract

Given graphs $H$ and $F$, the generalized Tur\'an number $\mathrm{ex}(n,H,F)$ is the largest number of copies of $H$ in $n$-vertex $F$-free graphs. We study the case when either $H$ or $F$ is a matching. We obtain several asymptotic and exact results.