Extremal $H$-free planar graphs

Abstract

Given a graph $H$, a graph is $H$-free if it does not contain $H$ as a subgraph. We continue to study the topic of "extremal" planar graphs, that is, how many edges can an $H$-free planar graph on $n$ vertices have? We define $e{x}_{{}_{\mathcal{P}}}(n,H)$ to be the maximum number of edges in an $H$-free planar graph on $n$ vertices. We first obtain several sufficient conditions on $H$ which yield $e{x}_{{}_{\mathcal{P}}}(n,H)=3n-6$ for all $n\ge |V(H)|$. We discover that the chromatic number of $H$ does not play a role, as in the celebrated Erdős-Stone Theorem. We then completely determine $e{x}_{{}_{\mathcal{P}}}(n,H)$ when $H$ is a wheel or a star. Finally, we examine the case when $H$ is a $(t,r)$-fan, that is, $H$ is isomorphic to $K_1+t{K}_{r-1}$, where $t\ge 2$ and $r\ge 3$ are integers. However, determining $e{x}_{{}_{\mathcal{P}}}(n,H)$, when $H$ is a planar subcubic graph, remains wide open.