A localized approach to generalized Tur\'an problems

TL;DR

This paper extends a localized approach to generalized Turán problems, providing tight bounds for the maximum weighted copies of a graph in $F$-free graphs, especially for graphs with dominating vertices.

Contribution

It introduces a weighted, localized method to analyze generalized Turán problems, yielding new asymptotic bounds for specific graph configurations.

Findings

Derived tight upper bounds for weighted copies of $H$ in $F$-free graphs.

Asymptotically determined ex$(n,H,K_{1,r})$ for graphs with dominating vertices.

Extended the localized approach to a broader class of Turán problems.

Abstract

Generalized Tur\'an problems ask for the maximum number of copies of a graph $H$ in an $n$-vertex, $F$-free graph, denoted by ex$(n,H,F)$. We show how to extend the new, localized approach of Brada\v{c}, Malec, and Tompkins to generalized Tur\'{a}n problems. We weight the copies of $H$ (typically taking $H=K_t$), instead of the edges, based on the size of the largest clique, path, or star containing the vertices of the copy of $H$, and in each case prove a tight upper bound on the sum of the weights. A consequence of our new localized theorems is an asymptotic determination of ex$(n,H,K_{1,r})$ for every $H$ having at least one dominating vertex and mex$(m,H,K_{1,r})$ for every $H$ having at least two dominating vertices.