Some stability and exact results in generalized Tur\'an problems

TL;DR

This paper investigates stability and exact solutions in generalized Turán problems, focusing on the maximum number of subgraphs in graphs avoiding a certain subgraph, and shows that near-extremal graphs resemble extremal configurations.

Contribution

The paper introduces new stability results and derives several exact solutions for generalized Turán problems, advancing understanding of extremal graph configurations.

Findings

New stability theorems for generalized Turán problems

Several exact extremal numbers determined

Characterization of near-extremal graphs

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. Stability refers to the usual phenomenon that if an $n$-vertex $F$-free graph $G$ contains almost $\mathrm{ex}(n,H,F)$ copies of $H$, than $G$ is in some sense similar to some extremal graph. We obtain new stability results for generalized Tur\'an problems and derive several new exact results.