Outerplanar Tur\'an number of a cycle

TL;DR

This paper provides a shorter proof for the maximum number of edges in an outerplanar graph with n vertices that avoids a cycle of length k, refining the understanding of extremal graph properties.

Contribution

The paper introduces a dual graph technique to establish a sharper upper bound for the outerplanar Turán number of cycles, simplifying previous proofs.

Findings

Established a shorter proof for the upper bound of $ex_ ext{OP}(n,C_k)$

Derived a precise upper bound formula for outerplanar Turán numbers of cycles

Enhanced the methodological approach using dual graph techniques

Abstract

A graph is outerplanar if it has a planar drawing for which all vertices belong to the outer face of the drawing. Let $H$ be a graph. The outerplanar Tur\'an number of $H$, denoted by $ex_\mathcal{OP}(n,H)$, is the maximum number of edges in an $n$-vertex outerplanar graph which does not contain $H$ as a subgraph. In 2021, L. Fang et al. determined the outerplanar Tur\'an number of cycles and paths. In this paper, we use techniques of dual graph to give a shorter proof for the sharp upperbound of $ex_\mathcal{OP}(n,C_k)\leq \frac{(2k - 5)(kn - k - 1)}{k^2 - 2k - 1}$.