Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Cycles in Sparse Graphs II   The {\em independence ratio} of a graph $G$ is defined by \[ \iota(G) :=
\sup_{X \subset V(G)} \frac{|X|}{\alpha(X)},\] where $\alpha(X)$ is the
independence number of the subgraph of $G$ induced by $X$. The independence
ratio is a relaxation of the chromatic number $\chi(G)$ in the sense that
$\chi(G) \geq \iota(G)$ for every graph $G$, while for many natural classes of
graphs these quantities are almost equal. In this paper, we address two old
conjectures of Erd\H{o}s on cycles in graphs with large chromatic number and a
conjecture of Erd\H{o}s and Hajnal on graphs with infinite chromatic number.

18
The categories of this paper are: econ
27
The categories of this paper are: comp-gas
8
The categories of this paper are: hep-ex
22
The categories of this paper are: chao-dyn
3
The categories of this paper are: physics
33
The categories of this paper are: plasm-ph
19
The categories of this paper are: eess
29
The categories of this paper are: funct-an
32
The categories of this paper are: atom-ph
35
The categories of this paper are: mtrl-th
7
The categories of this paper are: hep-th
31
The categories of this paper are: ao-sci
0
The categories of this paper are: hep-ph
15
The categories of this paper are: nucl-ex
4
The categories of this paper are: cond-mat
12
The categories of this paper are: nucl-th