Some exact results for non-degenerate generalized Tur\'an problems

TL;DR

This paper investigates exact values of the generalized Turán number for specific graph configurations, expanding known results beyond the extremal complete multipartite graphs.

Contribution

It provides new exact results for $ ext{ex}(n,H,F)$ with extremal graphs other than complete multipartite graphs, in the case where $ ext{chi}(H)< ext{chi}(F)$.

Findings

Exact values of $ ext{ex}(n,H,F)$ for new extremal graph classes

Extension of known results beyond complete multipartite extremal graphs

Results applicable when $ ext{chi}(H)< ext{chi}(F)$

Abstract

The generalized Tur\'an number $\mathrm{ex}(n,H,F)$ is the maximum number of copies of $H$ in $n$-vertex $F$-free graphs. We consider the case where $\chi(H)<\chi(F)$. There are several exact results on $\mathrm{ex}(n,H,F)$ when the extremal graph is a complete $(\chi(F)-1)$-partite graph. We obtain multiple exact results with other kinds of extremal graphs.