Outerplanar Tur\'{a}n numbers of cycles and paths

TL;DR

This paper determines the maximum number of edges in outerplanar graphs on n vertices that do not contain certain cycles or paths, providing exact Turán numbers for these graph classes.

Contribution

It completely characterizes the outerplanar Turán numbers for cycles and paths, filling a gap in extremal graph theory for outerplanar graphs.

Findings

Exact outerplanar Turán numbers for cycles are established.

Exact outerplanar Turán numbers for paths are established.

Results contribute to extremal graph theory for outerplanar graphs.

Abstract

A graph is outerplanar if it can be embedded in a plane such that all vertices lie on its outer face. The outerplanar Tur\'{a}n number of a given graph $H$, denoted by ${\rm ex}_{\mathcal{OP}}(n,H)$, is the maximum number of edges over all outerplanar graphs on $n$ vertices which do not contain a copy of $H$. In this paper, the outerplanar Tur\'{a}n numbers of cycles and paths are completely determined.