Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: On the number of non-isomorphic (simple) $k$-gonal biembeddings of
  complete multipartite graphs   This article aims to provide exponential lower bounds on the number of
non-isomorphic $k$-gonal biembeddings of the complete multipartite graph into
orientable surfaces.
  For this purpose, we use the concept, introduced by Archdeacon in 2015, of
Heffer array and its relations with graph embeddings. In particular we show
that, under certain hypotheses, from a single Heffter array, we can obtain an
exponential number of distinct graph embeddings. Exploiting this idea starting
from the arrays constructed by Cavenagh, Donovan and Yazici in 2020, we obtain
that, for infinitely many values of $k$ and $v$, there are at least
$k^{\frac{k}{2}+o(k)} \cdot 2^{v\cdot \frac{H(1/4)}{(2k)^2}+o(v)}$
non-isomorphic $k$-gonal biembeddings of $K_v$, where $H(\cdot)$ is the binary
entropy. Moreover about the embeddings of $K_{\frac{v}{t}\times t}$, for
$t\in\{1,2,k\}$, we provide a construction of $2^{v\cdot
\frac{H(1/4)}{2k(k-1)}+o(v,k)}$ non-isomorphic $k$-gonal biembeddings whenever
$k$ is odd and $v$ belongs to a wide infinite family of values.

34
The categories of this paper are: chem-ph
32
The categories of this paper are: atom-ph
11
The categories of this paper are: hep-lat
6
The categories of this paper are: astro-ph
36
The categories of this paper are: cmp-lg
4
The categories of this paper are: cond-mat
31
The categories of this paper are: ao-sci
18
The categories of this paper are: econ
16
The categories of this paper are: stat
5
The categories of this paper are: gr-qc
17
The categories of this paper are: q-fin
22
The categories of this paper are: chao-dyn
10
The categories of this paper are: q-bio
21
The categories of this paper are: adap-org
37
The categories of this paper are: supr-con
33
The categories of this paper are: plasm-ph