Generalized outerplanar Tur\'an number of short paths

TL;DR

This paper determines the maximum number of certain paths within outerplanar graphs, providing exact and asymptotic values for paths of length 4 and 5, respectively, and characterizes extremal graphs.

Contribution

It offers the first exact value for the maximum copies of P4 and asymptotic for P5 in outerplanar graphs, along with a characterization of extremal graphs.

Findings

Exact value of $f_{OP}(n, P_4)$ established.

Asymptotic value of $f_{OP}(n, P_5)$ determined.

Characterization of all extremal outerplanar graphs for $P_4$.

Abstract

Let $H$ be a graph. The generalized outerplanar Tur\'an number of $H$, denoted by $f_{\mathcal{OP}}(n,H)$, is the maximum number of copies of $H$ in an $n$-vertex outerplanar graph. Let $P_k$ be the path on $k$ vertices. In this paper we give an exact value of $f_{\mathcal{OP}}(n,P_4)$ and a best asymptotic value of $f_{\mathcal{OP}}(n,P_5)$. Moreover, we characterize all outerplanar graphs containing $f_{\mathcal{OP}}(n,P_4)$ copies of $P_4$.