Planar Turán Number of Double Stars

Abstract

Given a graph $F$, the planar Turán number of $F$, denoted ${\text{ex}}_{\mathcal{P}}(n,F)$, is the maximum number of edges in an $n$-vertex $F$-free planar graph. Such an extremal graph problem was initiated by Dowden while determining sharp upper bound for ${\text{ex}}_{\mathcal{P}}(n,C_4)$ and ${\text{ex}}_{\mathcal{P}}(n,C_5)$, where $C_4$ and $C_5$ are cycles of length four and five respectively. In this paper we determined an upper bound for ${\text{ex}}_{\mathcal{P}}(n,S_{2,2})$, ${\text{ex}}_{\mathcal{P}}(n,S_{2,3})$, ${\text{ex}}_{\mathcal{P}}(n,S_{2,4})$, ${\text{ex}}_{\mathcal{P}}(n,S_{2,5})$, ${\text{ex}}_{\mathcal{P}}(n,S_{3,3})$ and ${\text{ex}}_{\mathcal{P}}(n,S_{3,4})$, where $S_{m,n}$ is a double star with $m$ and $n$ leafs. Moreover, the bounds for ${\text{ex}}_{\mathcal{P}}(n,S_{2,2})$ and ${\text{ex}}_{\mathcal{P}}(n,S_{2,3})$ are sharp.