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 $ex_{_\mathcal{P}}(n,\mathcal{F})$ to be the maximum number of edges in an $\mathcal{F}$-free planar graph on $n$ vertices. Dowden obtained the tight bounds $ex_{_\mathcal{P}}(n,C_4)\leq15(n-2)/7$ for all $n\geq4$ and $ex_{_\mathcal{P}}(n,C_5)\leq(12n-33)/5$ for all $n\geq11$. In this paper, we continue to promote the idea of determining $ex_{_\mathcal{P}}(n,\mathcal{F})$ for certain classes $\mathcal{F}$. Let $\Theta_k$ denote the family of Theta graphs on $k\ge4$ vertices, that is, graphs obtained from a cycle $C_k$ by adding an additional edge joining two non-consecutive vertices. The study of $ex_{_\mathcal{P}}(n,\Theta_4)$ was suggested by Dowden. We show that $ex_{_\mathcal{P}}(n,\Theta_4)\leq12(n-2)/5$ for all $n\geq 4$, $ex_{_\mathcal{P}}(n,\Theta_5)\leq5(n-2)/2$ for all $n\ge5$, 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 $ex_{_\mathcal{P}}(n,C_6)\le ex_{_\mathcal{P}}(n,\Theta_6)\le 18(n-2)/7$ for all $n\ge6$.