Turán type problems for a fixed graph and a linear forest

Abstract

Let $\mathcal{F}$ be a family of graphs. A graph $G$ is $\mathcal{F}$-free if $G$ does not contain any $F\in \mathcal{F}$ as a subgraph. The Turán number, denoted by $ex(n,\mathcal{F})$, is the maximum number of edges in an $n$-vertex $\mathcal{F}$-free graph. Let $F$ be a fixed graph with $\ddot{I}‡(F)\ge 3$. A forest $H$ is called a linear forest if all components of $H$ are paths. In this paper, we determined the exact value of $ex(n,\{H,F\})$ for a fixed graph $F$ with $\ddot{I}‡(F)\ge 3$ and a linear forest $H$ with at least $2$ components and each component with size at least $3$.