Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Sparse halves in $K_4$-free graphs   A conjecture of Chung and Graham states that every $K_4$-free graph on $n$
vertices contains a vertex set of size $\lfloor n/2 \rfloor$ that spans at most
$n^2/18$ edges. We make the first step toward this conjecture by showing that
it holds for all regular graphs.

23
The categories of this paper are: solv-int
36
The categories of this paper are: cmp-lg
31
The categories of this paper are: ao-sci
26
The categories of this paper are: bayes-an
1
The categories of this paper are: cs
28
The categories of this paper are: alg-geom
13
The categories of this paper are: quant-ph
33
The categories of this paper are: plasm-ph
12
The categories of this paper are: nucl-th
4
The categories of this paper are: cond-mat
8
The categories of this paper are: hep-ex
0
The categories of this paper are: hep-ph
24
The categories of this paper are: dg-ga
14
The categories of this paper are: math-ph
30
The categories of this paper are: q-alg
34
The categories of this paper are: chem-ph