Abstract: An H-free graph is a graph not containing the given graph H as a subgraph. It is well known that the Turán number $ \mathrm{{ex}}_{}(n,H) $ is the maximum number of edges in an H-free graph on n vertices. Based on this definition, we define $ \mathrm{{ex}}_{_\mathcal {P}}(n,H) $ to restrict the graph classes to planar graphs, that is, $ \mathrm{{ex}}_{_\mathcal {P}}(n,H)=\max \{|E(G)|: G\in {\mathcal {G}} $, where $ {\mathcal {G}} $ is a family of all H-free planar graphs on n vertices. Obviously, we have $ \mathrm{{ex}}_{_{\mathcal {P}}}(n,H)=3n-6 $ if the graph H is not a planar graph. The study is initiated by Dowden (J Graph Theory 83:213–230, 2016), who obtained some results when H is considered as $ C_4 $ or $ C_5 $. In this paper, we determine the values of $ \mathrm{{ex}}_{_{\mathcal {P}}}(n,P_k) $ with $ k\in \{8,9\} $, where $ P_k $ is a path with k vertices.

Key words: Turán number, Extremal graph, Planar graph