Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: The maximum number of 10- and 12-cycles in a planar graph   For a fixed planar graph $H$, let
$\operatorname{\mathbf{N}}_{\mathcal{P}}(n,H)$ denote the maximum number of
copies of $H$ in an $n$-vertex planar graph. In the case when $H$ is a cycle,
the asymptotic value of $\operatorname{\mathbf{N}}_{\mathcal{P}}(n,C_m)$ is
currently known for $m\in\{3,4,5,6,8\}$. In this note, we extend this list by
establishing $\operatorname{\mathbf{N}}_{\mathcal{P}}(n,C_{10})\sim(n/5)^5$ and
$\operatorname{\mathbf{N}}_{\mathcal{P}}(n,C_{12})\sim(n/6)^6$. We prove this
by answering the following question for $m\in\{5,6\}$, which is interesting in
its own right: which probability mass $\mu$ on the edges of some clique
maximizes the probability that $m$ independent samples from $\mu$ form an
$m$-cycle?

10
The categories of this paper are: q-bio
3
The categories of this paper are: physics
16
The categories of this paper are: stat
11
The categories of this paper are: hep-lat
31
The categories of this paper are: ao-sci
24
The categories of this paper are: dg-ga
5
The categories of this paper are: gr-qc
0
The categories of this paper are: hep-ph
22
The categories of this paper are: chao-dyn
15
The categories of this paper are: nucl-ex
27
The categories of this paper are: comp-gas
8
The categories of this paper are: hep-ex
29
The categories of this paper are: funct-an
4
The categories of this paper are: cond-mat
32
The categories of this paper are: atom-ph
37
The categories of this paper are: supr-con