Planar Tur\'an number for balanced double stars

TL;DR

This paper determines the exact maximum number of edges in large planar graphs that do not contain a specific balanced double star, completing the understanding for all such double stars.

Contribution

It provides the exact planar Turán number for the balanced double star $S_{3,3}$, resolving a key open case in the study of planar Turán numbers.

Findings

Exact value of $ex_{ ext{planar}}(n, S_{3,3})$ established

Complete characterization of planar Turán numbers for all balanced double stars

Advances understanding of extremal planar graphs avoiding double star subgraphs

Abstract

Planar Tur\'an number, denoted by $ex_{\mathcal{P}}(n,H)$, is the maximum number of edges in an $n$-vertex planar graph which does not contain $H$ as a subgraph. Ghosh, Gy\H{o}ri, Paulos and Xiao initiated the topic of the planar Tur\'an number for double stars. For balanced double star, $S_{3,3}$ is the only remaining graph need to be considered. In this paper, we give the exact value of $ex_{\mathcal{P}}(n,S_{3,3})$, forcing the planar Tur\'an number for all balanced double stars completely determined.