Generalized Tur\'{a}n number for linear forests

TL;DR

This paper determines the maximum number of complete subgraphs in large, linear forest-free graphs, extending classical Turán number results and characterizing extremal structures.

Contribution

It provides exact values for the generalized Turán number for linear forests with large minimum degree, improving previous bounds for specific cases.

Findings

Determined $ex(n,K_s,F)$ for large $n$ and certain linear forests.

Characterized the structure of extremal $F$-free graphs with large minimum degree.

Improved classical Turán number bounds for linear forests when $s=2$.

Abstract

The generalized Tur\'{a}n number $ex(n,K_s,H)$ is defined to be the maximum number of copies of a complete graph $K_s$ in any $H$-free graph on $n$ vertices. Let $F$ be a linear forest consisting of $k$ paths of orders $\ell_1,\ell_2,...,\ell_k$. In this paper, by characterizing the structure of the $F$-free graph with large minimum degree, we determine the value of $ex(n,K_s,F)$ for $n=\Omega\left(|F|^s\right)$ and $k\geq 2$ except some $\ell_i=3$, and the corresponding extremal graphs. The special case when $s=2$ of our result improves some results of Bushaw and Kettle (2011) and Lidick\'{y} et al. (2013) on the classical Tur\'{a}n number for linear forests.