The number of induced paths in outerplanar graphs

Let $P_k$ denote the path with $k$ vertices, and ${\mathrm{e}\mathrm{x}}_{\mathcal{O}\mathcal{P}}(n,H^{\mathrm{i}\mathrm{n}\mathrm{d}},\mathrm{\varnothing })$ be the maximum number of induced copies of $H$ in an $n$-vertex outerplanar graph. In this paper, we determine the exact value of ${\mathrm{e}\mathrm{x}}_{\mathcal{O}\mathcal{P}}(n,P_3^{\mathrm{i}\mathrm{n}\mathrm{d}},\mathrm{\varnothing })$ for all $n$, and give an asymptotic value of ${\mathrm{e}\mathrm{x}}_{\mathcal{O}\mathcal{P}}(n,P_4^{\mathrm{i}\mathrm{n}\mathrm{d}},\mathrm{\varnothing })$. For general $k$, Matolcsi and Nagy proved that ${\lim }_{k\to \mathrm{\infty }}{({\mathrm{e}\mathrm{x}}_{\mathcal{O}\mathcal{P}}(n,P_{k+1},\mathrm{\varnothing }))}^{1\mathrm{/}k}=4$. In the induced case, we prove that $$fib(k-1)\frac{{(n-2k+3)}^2}{4}\le {\mathrm{e}\mathrm{x}}_{\mathcal{O}\mathcal{P}}(n,P_{k+1}^{\mathrm{i}\mathrm{n}\mathrm{d}},\mathrm{\varnothing })\le fib(k+1)\left(\genfrac{}{}{0px}{}{n}{2}\right),$$ where $fib(k)$ is the Fibonacci number. This implies that ${\lim }_{k\to \mathrm{\infty }}{({\mathrm{e}\mathrm{x}}_{\mathcal{O}\mathcal{P}}(n,P_{k+1}^{\mathrm{i}\mathrm{n}\mathrm{d}},\mathrm{\varnothing }))}^{1\mathrm{/}k}=\frac{\sqrt{5}+1}{2}\approx 1.618$.