The maximum number of $P_\ell$ copies in $P_k$-free graphs

TL;DR

This paper investigates the maximum number of path copies in $P_k$-free graphs, providing asymptotic and exact results that extend classical extremal theorems to specific graph structures.

Contribution

It offers new asymptotic and exact bounds for the number of paths in $P_k$-free graphs, extending Erdős-Gallai type theorems to these cases.

Findings

Derived asymptotically sharp estimates for paths, stars, and cycles in $H$-free graphs.

Extended classical extremal theorems to new graph structures.

Provided exact results in specific cases for maximum path copies.

Abstract

Generalizing Tur\'an's classical extremal problem, Alon and Shikhelman investigated the problem of maximizing the number of $T$ copies in an $H$-free graph, for a pair of graphs $T$ and $H$. Whereas Alon and Shikhelman were primarily interested in determining the order of magnitude for large classes of graphs $H$, we focus on the case when $T$ and $H$ are paths, where we find asymptotic and in some cases exact results. We also consider other structures like stars and the set of cycles of length at least $k$, where we derive asymptotically sharp estimates. Our results generalize well-known extremal theorems of Erd\H{o}s and Gallai.