Planar Tur\'an number for balanced double stars

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.