Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Planar Tur\'an Number of the $\Theta_6$   Let $F$ be a graph. The planar Tur\'an number of $F$, denoted by
$\text{ex}_{\mathcal{P}}(n,F)$, is the maximum number of edges in an $n$-vertex
planar graph containing no copy of $F$ as a subgraph. Let $\Theta_k$ denote the
family of Theta graphs on $k\geq 4$ vertices, that is, a graph obtained by
joining a pair of non-consecutive vertices of a $k$-cycle with an edge. Y. Lan,
et.al. determined sharp upper bound for $\text{ex}_{\mathcal{P}}(n,\Theta_4)$
and $\text{ex}_{\mathcal{P}}(n,\Theta_5)$. Moreover, they obtained an upper
bound for $\text{ex}_{\mathcal{P}}(n,\Theta_6)$. They proved that,
$\text{ex}_{\mathcal{P}}(n,\Theta_6)\leq \frac{18}{7}n-\frac{36}{7}$. In this
paper, we improve their result by giving a bound which is sharp. In particular,
we prove that $\text{ex}_{\mathcal{P}}(n,\Theta_6)\leq
\frac{18}{7}n-\frac{48}{7}$ and demonstrate that there are infinitely many $n$
for which there exists a $\Theta_6$-free planar graph $G$ on $n$ vertices,
which attains the bound.

17
The categories of this paper are: q-fin
19
The categories of this paper are: eess
1
The categories of this paper are: cs
7
The categories of this paper are: hep-th
29
The categories of this paper are: funct-an
24
The categories of this paper are: dg-ga
25
The categories of this paper are: patt-sol
22
The categories of this paper are: chao-dyn
20
The categories of this paper are: acc-phys
26
The categories of this paper are: bayes-an
35
The categories of this paper are: mtrl-th
23
The categories of this paper are: solv-int
16
The categories of this paper are: stat
32
The categories of this paper are: atom-ph
15
The categories of this paper are: nucl-ex
13
The categories of this paper are: quant-ph