The Turán number of path-star forests

The Turán number of a graph $H$, denoted by $ex(n,H)$, is the maximum number of edges in any graph on $n$ vertices containing no $H$ as a subgraph. A linear (star) forest is a forest consisting of paths (stars). A path-star forest $F$ is a forest consisting of paths and stars. In this paper, we determine $ex(n,F)$ for sufficiently large $n$ and characterize the corresponding extremal graphs, and our result generalizes previous known results on the Turán numbers of linear forests and star forests.