Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Dichromatic number and fractional chromatic number   The dichromatic number of a graph $G$ is the maximum integer $k$ such that
there exists an orientation of the edges of $G$ such that for every partition
of the vertices into fewer than $k$ parts, at least one of the parts must
contain a directed cycle under this orientation. In 1979, Erd\H{o}s and
Neumann-Lara conjectured that if the dichromatic number of a graph is bounded,
so is its chromatic number. We make the first significant progress on this
conjecture by proving a fractional version of the conjecture. While our result
uses stronger assumption about the fractional chromatic number, it also gives a
much stronger conclusion: If the fractional chromatic number of a graph is at
least $t$, then the fractional version of the dichromatic number of the graph
is at least $\tfrac{1}{4}t/\log_2(2et^2)$. This bound is best possible up to a
small constant factor. Several related results of independent interest are
given.

20
The categories of this paper are: acc-phys
25
The categories of this paper are: patt-sol
15
The categories of this paper are: nucl-ex
35
The categories of this paper are: mtrl-th
7
The categories of this paper are: hep-th
18
The categories of this paper are: econ
10
The categories of this paper are: q-bio
8
The categories of this paper are: hep-ex
6
The categories of this paper are: astro-ph
11
The categories of this paper are: hep-lat
34
The categories of this paper are: chem-ph
29
The categories of this paper are: funct-an
0
The categories of this paper are: hep-ph
3
The categories of this paper are: physics
33
The categories of this paper are: plasm-ph
26
The categories of this paper are: bayes-an