Planar Turán Number of the ${\Theta }_6$

Abstract

Let $\mathcal{F}$ be a nonempty family of graphs. A graph $G$ is called $\mathcal{F}$-\textit{free} if it contains no graph from $\mathcal{F}$ as a subgraph. For a positive integer $n$, the \emph{planar Turán number} of $\F$, denoted by $\ex_{\p}(n,\F)$, is the maximum number of edges in an $n$-vertex $\F$-free planar graph. Let ${\Theta }_k$ be the family of Theta graphs on $k\ge 4$ vertices, that is, graphs obtained by joining a pair of non-consecutive vertices of a $k$-cycle with an edge. Lan, Shi and Song determined an upper bound ${\text{ex}}_{\mathcal{P}}(n,{\Theta }_6)\le \frac{18}{7}n-\frac{36}{7}$, but for large $n$, they did not verify that the bound is sharp. In this paper, we improve their bound by proving ${\text{ex}}_{\mathcal{P}}(n,{\Theta }_6)\le \frac{18}{7}n-\frac{48}{7}$ and then we demonstrate the existence of infinitely many positive integer $n$ and an $n$-vertex ${\Theta }_6$-free planar graph attaining the bound.

arXiv comment27 pages, 21 figures