For a planar graph $H$, let $\operatorname{ \mathbf{ N}}_{ \mathcal P}(n,H)$ denote the maximum number of copies of $H$ in an $n$-vertex planar graph. In this paper, we prove that $\operatorname{ \mathbf{ N}}_{ \mathcal P}(n,P_7)\sim{ 4\over 27}n^4$, $\operatorname{ \mathbf{ N}}_{ \mathcal P}(n,C_6)\sim(n/3)^3$, $\operatorname{ \mathbf{ N}}_{ \mathcal P}(n,C_8)\sim(n/4)^4$ and $\operatorname{ \mathbf{ N}}_{ \mathcal P}(n,K_4\{ 1\})\sim(n/6)^6$, where $K_4\{ 1\}$ is the $1$-subdivision of $K_4$. In addition, we obtain significantly improved upper bounds on $\operatorname{ \mathbf{ N}}_{ \mathcal P}(n,P_{ 2m+1})$ and $\operatorname{ \mathbf{ N}}_{ \mathcal P}(n,C_{ 2m})$ for $m\geq 4$. For a wide class of graphs $H$, the key technique developed in this paper allows us to bound $\operatorname{ \mathbf{ N}}_{ \mathcal P}(n,H)$ in terms of an optimization problem over weighted graphs.
Categories: math.CO