Extremal $P_8$-Free/$P_9$-Free Planar Graphs

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.