On generalized Tur\'an problems with bounded matching number

TL;DR

This paper investigates generalized Turán problems with bounded matching number, establishing stability results and exact values for specific cases, advancing understanding of extremal graph configurations under these constraints.

Contribution

It introduces stability results for generalized Turán problems with bounded matching number and determines exact extremal numbers for certain non-bipartite graphs and paths.

Findings

Established stability results for the problems.

Provided exact extremal numbers for specific graph classes.

Extended understanding of Turán problems with bounded matching constraints.

Abstract

The generalized Tur\'an number $\mathrm{ex}(n, H, \mathcal{F})$ is defined as the maximum number of copies of a graph $H$ in an $n$-vertex graph that does not contain any graph $F \in \mathcal{F}$. Alon and Frankl initiated the study of Tur\'an problems with a bounded matching number.In this paper, we establish stability results for generalized Tur\'an problems with bounded matching number.Using the stability results, we provide exact values of $\ex(n,K_r,\{F,M_{s+1}\})$ for $F$ being any non-bipartite graph or a path on $k$ vertices.