On Tur\'an-good graphs

TL;DR

This paper investigates the maximum number of copies of a graph H in an F-free graph, establishing general theorems for when F has certain chromatic properties and precisely determining extremal numbers for specific graph pairs.

Contribution

It provides a general theorem for F with an edge whose removal lowers chromatic number, and determines exact extremal numbers for certain cycles and paths.

Findings

Determines $ex(n,P_k,C_{2 ext{l}+1})$ exactly for large n.

Determines $ex(n,C_{2k},C_{2 ext{l}+1})$ exactly for large n.

Establishes a general theorem for F with an edge decreasing chromatic number.

Abstract

For graphs $H$ and $F$, the generalized Tur\'an number $ex(n,H,F)$ is the largest number of copies of $H$ in an $F$-free graph on $n$ vertices. We say that $H$ is $F$-Tur\'an-good if $ex(n,H,F)$ is the number of copies in the $(\chi(F)-1)$-partite Tur\'an graph, provided $n$ is large enough. We present a general theorem in case $F$ has an edge whose deletion decreases the chromatic number. In particular, this determines $ex(n,P_k,C_{2\ell+1})$ and $ex(n,C_{2k},C_{2\ell+1})$ exactly, if $n$ is large enough. We also study the case when $F$ has a vertex whose deletion decreases the chromatic number.