An improved upper bound for planar Turán number of double star $S_{2,5}$

Abstract

The planar Turán number of a graph $H$, denoted by $e{x}_{\mathcal{P}}(n,H)$, is the maximum number of edges in an $n$-vertex $H$-free planar graph. Recently, D. Ghosh, et al. initiated the topic of double stars and prove that $e{x}_{\mathcal{P}}(n,S_{2,5})\le \frac{20}{7}n$. In this paper, we continue to study this and give a sharp upper bound $e{x}_{\mathcal{P}}(n,S_{2,5})\le \frac{19}{7}n-\frac{18}{7}$ for all $n\ge 1$, with equality when $n=12$. This improves Ghosh's result.