The number of induced paths in outerplanar graphs

TL;DR

This paper determines the maximum number of induced paths of fixed length in outerplanar graphs, providing exact and asymptotic results, and establishing growth limits related to Fibonacci numbers.

Contribution

It exactly computes the maximum induced 3-paths and asymptotically estimates longer paths, extending known results with Fibonacci-based bounds.

Findings

Exact value for induced 3-paths in outerplanar graphs.

Asymptotic bounds for longer induced paths.

Growth rate limit related to the golden ratio.

Abstract

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