Extremal Theta-free planar graphs

Given a family $\mathcal{F}$, a graph is $\mathcal{F}$-free if it does not contain any graph in $\mathcal{F}$ as a subgraph. We study the topic of “extremal” planar graphs initiated by Dowden [J. Graph Theory 83 (2016) 213–230], that is, how many edges can an $\mathcal{F}$-free planar graph on $n$ vertices have? We define $e{x}_{{}_{\mathcal{P}}}(n,\mathcal{F})$ to be the maximum number of edges in an $\mathcal{F}$-free planar graph on $n$ vertices. Dowden obtained the tight bounds $e{x}_{{}_{\mathcal{P}}}(n,C_4)\le 15(n-2)\mathrm{/}7$ for all $n\ge 4$ and $e{x}_{{}_{\mathcal{P}}}(n,C_5)\le (12n-33)\mathrm{/}5$ for all $n\ge 11$. In this paper, we continue to promote the idea of determining $e{x}_{{}_{\mathcal{P}}}(n,\mathcal{F})$ for certain classes $\mathcal{F}$. Let ${\mathrm{\Theta }}_k$ denote the family of Theta graphs on $k\ge 4$ vertices, that is, graphs obtained from a cycle $C_k$ by adding an additional edge joining two non-consecutive vertices. The study of $e{x}_{{}_{\mathcal{P}}}(n,{\mathrm{\Theta }}_4)$ was suggested by Dowden. We show that $e{x}_{{}_{\mathcal{P}}}(n,{\mathrm{\Theta }}_4)\le 12(n-2)\mathrm{/}5$ for all $n\ge 4$, $e{x}_{{}_{\mathcal{P}}}(n,{\mathrm{\Theta }}_5)\le 5(n-2)\mathrm{/}2$ for all $n\ge 5$, and then demonstrate that these bounds are tight, in the sense that there are infinitely many values of $n$ for which they are attained exactly. We also prove that $ex_{$\mathcal{P}$}(n,C_6)\le ex{_$\mathcal{P}$}(n,\Theta_6)\le 18(n-2)/7$forall$n\ge6$.