Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: A note on the orientation covering number   Given a graph $G$, its orientation covering number $\sigma(G)$ is the
smallest non-negative integer $k$ with the property that we can choose $k$
orientations of $G$ such that whenever $x, y, z$ are vertices of $G$ with
$xy,xz\in E(G)$ then there is a chosen orientation in which both $xy$ and $xz$
are oriented away from $x$. Esperet, Gimbel and King showed that $\sigma(G)\leq
\sigma\left(K_{\chi(G)}\right)$, where $\chi(G)$ is the chromatic number of
$G$, and asked whether we always have equality. In this note we prove that it
is indeed always the case that $\sigma(G)=\sigma(K_{\chi(G)})$. We also
determine the exact value of $\sigma(K_n)$ explicitly for `most' values of $n$.

13
The categories of this paper are: quant-ph
32
The categories of this paper are: atom-ph
9
The categories of this paper are: nlin
18
The categories of this paper are: econ
22
The categories of this paper are: chao-dyn
27
The categories of this paper are: comp-gas
34
The categories of this paper are: chem-ph
19
The categories of this paper are: eess
21
The categories of this paper are: adap-org
17
The categories of this paper are: q-fin
10
The categories of this paper are: q-bio
37
The categories of this paper are: supr-con
35
The categories of this paper are: mtrl-th
14
The categories of this paper are: math-ph
1
The categories of this paper are: cs
5
The categories of this paper are: gr-qc