The generalized Tur\'{a}n number of spanning linear forests

TL;DR

This paper determines the exact maximum number of certain subgraphs in graphs avoiding linear forests, extending to bipartite graphs, using the shifting method to solve generalized Turán problems.

Contribution

It provides exact values for generalized Turán numbers involving complete and modified bipartite graphs in linear forest-free graphs, including bipartite cases.

Findings

Exact values of $ex(n,K_s, ext{linear forests})$ and $ex(n,K^*_{s,t}, ext{linear forests})$

Determination of $ex_{bip}(n,K_{s,t}, ext{linear forests})$ in bipartite graphs

Application of shifting method to solve Turán-type extremal problems

Abstract

Let $\mathcal{F}$ be a family of graphs. A graph $G$ is called \textit{$\mathcal{F}$-free} if for any $F\in \mathcal{F}$, there is no subgraph of $G$ isomorphic to $F$. Given a graph $T$ and a family of graphs $\mathcal{F}$, the generalized Tur\'{a}n number of $\mathcal{F}$ is the maximum number of copies of $T$ in an $\mathcal{F}$-free graph on $n$ vertices, denoted by $ex(n,T,\mathcal{F})$. A linear forest is a graph whose connected components are all paths or isolated vertices. Let $\mathcal{L}_{n,k}$ be the family of all linear forests of order $n$ with $k$ edges and $K^*_{s,t}$ a graph obtained from $K_{s,t}$ by substituting the part of size $s$ with a clique of the same size. In this paper, we determine the exact values of $ex(n,K_s,\mathcal{L}_{n,k})$ and $ex(n,K^*_{s,t},\mathcal{L}_{n,k})$. Also, we study the case of this problem when the \textit{"host graph"} is bipartite. Denote by $ex_{bip}(n,T,\mathcal{F})$ the maximum possible number of copies of $T$ in an $\mathcal{F}$-free bipartite graph with each part of size $n$. We determine the exact value of $ex_{bip}(n,K_{s,t},\mathcal{L}_{n,k})$. Our proof is mainly based on the shifting method.