Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: On representable graphs, semi-transitive orientations, and the
  representation numbers   A graph $G=(V,E)$ is representable if there exists a word $W$ over the
alphabet $V$ such that letters $x$ and $y$ alternate in $W$ if and only if
$(x,y)\in E$ for each $x\neq y$. If $W$ is $k$-uniform (each letter of $W$
occurs exactly $k$ times in it) then $G$ is called $k$-representable. It was
shown that a graph is representable if and only if it is $k$-representable for
some $k$. Minimum $k$ for which a representable graph $G$ is $k$-representable
is called its representation number.
  In this paper we give a characterization of representable graphs in terms of
orientations. Namely, we show that a graph is representable if and only if it
admits an orientation into a so-called \emph{semi-transitive digraph}. This
allows us to prove a number of results about representable graphs, not the
least that 3-colorable graphs are representable. We also prove that the
representation number of a graph on $n$ nodes is at most $n$, from which one
concludes that the recognition problem for representable graphs is in NP. This
bound is tight up to a constant factor, as we present a graph whose
representation number is $n/2$.
  We also answer several questions, in particular, on representability of the
Petersen graph and local permutation representability.

29
The categories of this paper are: funct-an
21
The categories of this paper are: adap-org
35
The categories of this paper are: mtrl-th
34
The categories of this paper are: chem-ph
20
The categories of this paper are: acc-phys
24
The categories of this paper are: dg-ga
1
The categories of this paper are: cs
32
The categories of this paper are: atom-ph
3
The categories of this paper are: physics
23
The categories of this paper are: solv-int
18
The categories of this paper are: econ
31
The categories of this paper are: ao-sci
6
The categories of this paper are: astro-ph
11
The categories of this paper are: hep-lat
8
The categories of this paper are: hep-ex
14
The categories of this paper are: math-ph