Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Counting orientations of graphs with no strongly connected tournaments   Let $S_k(n)$ be the maximum number of orientations of an $n$-vertex graph $G$
in which no copy of $K_k$ is strongly connected. For all integers $n$, $k\geq
4$ where $n\geq 5$ or $k\geq 5$, we prove that $S_k(n) = 2^{t_{k-1}(n)}$, where
$t_{k-1}(n)$ is the number of edges of the $n$-vertex $(k-1)$-partite Tur\'an
graph $T_{k-1}(n)$, and that $T_{k-1}(n)$ is the only $n$-vertex graph with
this number of orientations. Furthermore, $S_4(4) = 40$ and this maximality is
achieved only by $K_4$.

28
The categories of this paper are: alg-geom
34
The categories of this paper are: chem-ph
10
The categories of this paper are: q-bio
8
The categories of this paper are: hep-ex
4
The categories of this paper are: cond-mat
0
The categories of this paper are: hep-ph
15
The categories of this paper are: nucl-ex
19
The categories of this paper are: eess
12
The categories of this paper are: nucl-th
18
The categories of this paper are: econ
35
The categories of this paper are: mtrl-th
20
The categories of this paper are: acc-phys
31
The categories of this paper are: ao-sci
7
The categories of this paper are: hep-th
24
The categories of this paper are: dg-ga
11
The categories of this paper are: hep-lat