Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Planar Tur\'an number of the 6-cycle   Let ${\rm ex}_{\mathcal{P}}(n,T,H)$ denote the maximum number of copies of
$T$ in an $n$-vertex planar graph which does not contain $H$ as a subgraph.
When $T=K_2$, ${\rm ex}_{\mathcal{P}}(n,T,H)$ is the well studied function, the
planar Tur\'an number of $H$, denoted by ${\rm ex}_{\mathcal{P}}(n,H)$. The
topic of extremal planar graphs was initiated by Dowden (2016). He obtained
sharp upper bound for both ${\rm ex}_{\mathcal{P}}(n,C_4)$ and ${\rm
ex}_{\mathcal{P}}(n,C_5)$. Later on, Y. Lan, et al. continued this topic and
proved that ${\rm ex}_{\mathcal{P}}(n,C_6)\leq \frac{18(n-2)}{7}$. In this
paper, we give a sharp upper bound ${\rm ex}_{\mathcal{P}}(n,C_6) \leq
\frac{5}{2}n-7$, for all $n\geq 18$, which improves Lan's result. We also pose
a conjecture on ${\rm ex}_{\mathcal{P}}(n,C_k)$, for $k\geq 7$.

20
The categories of this paper are: acc-phys
37
The categories of this paper are: supr-con
0
The categories of this paper are: hep-ph
13
The categories of this paper are: quant-ph
17
The categories of this paper are: q-fin
1
The categories of this paper are: cs
3
The categories of this paper are: physics
10
The categories of this paper are: q-bio
21
The categories of this paper are: adap-org
25
The categories of this paper are: patt-sol
35
The categories of this paper are: mtrl-th
15
The categories of this paper are: nucl-ex
24
The categories of this paper are: dg-ga
12
The categories of this paper are: nucl-th
16
The categories of this paper are: stat
7
The categories of this paper are: hep-th