Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: A density bound for triangle-free $4$-critical graphs   We show every triangle-free $4$-critical graph $G$ satisfies $e(G) \geq
\frac{5v(G)+2}{3}$. In fact, we prove a more general result characterizing
sparse 4-critical graphs with few vertex- disjoint triangles.

8
The categories of this paper are: hep-ex
20
The categories of this paper are: acc-phys
21
The categories of this paper are: adap-org
25
The categories of this paper are: patt-sol
28
The categories of this paper are: alg-geom
31
The categories of this paper are: ao-sci
33
The categories of this paper are: plasm-ph
7
The categories of this paper are: hep-th
12
The categories of this paper are: nucl-th
24
The categories of this paper are: dg-ga
37
The categories of this paper are: supr-con
3
The categories of this paper are: physics
18
The categories of this paper are: econ
0
The categories of this paper are: hep-ph
14
The categories of this paper are: math-ph
17
The categories of this paper are: q-fin