Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Improved bounds on the chromatic numbers of the square of Kneser graphs   The Kneser graph $K(n,k)$ is the graph whose vertices are the $k$-elements
subsets of an $n$-element set, with two vertices adjacent if the sets are
disjoint. The square $G^2$ of a graph $G$ is the graph defined on $V(G)$ such
that two vertices $u$ and $v$ are adjacent in $G^2$ if the distance between $u$
and $v$ in $G$ is at most 2. Determining the chromatic number of the square of
the Kneser graph $K(2k+1, k)$ is an interesting problem, but not much progress
has been made. Kim and Nakprasit \cite{2004KN} showed that $\chi(K^2(2k+1,k))
\leq 4k+2$, and Chen, Lih, and Wu \cite{2009CLW} showed that $\chi(K^2(2k+1,k))
\leq 3k+2$ for $k \geq 3$. In this paper, we give improved upper bounds on
$\chi(K^2(2k+1,k))$. We show that $\chi(K^2(2k+1,k)) \leq 2k+2$, if $ 2k +1 =
2^n -1$ for some positive integer $n$. Also we show that $\chi(K^2(2k+1,k))
\leq \frac{8}{3}k+\frac{20}{3}$ for every integer $k\ge 2$. In addition to
giving improved upper bounds, our proof is concise and can be easily understood
by readers while the proof in \cite{2009CLW} is very complicated. Moreover, we
show that $\chi(K^2(2k+r,k))=\Theta(k^r)$ for each integer $2 \leq r \leq k-2$.

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The categories of this paper are: astro-ph
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The categories of this paper are: acc-phys
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The categories of this paper are: ao-sci
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The categories of this paper are: bayes-an
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The categories of this paper are: alg-geom
32
The categories of this paper are: atom-ph
30
The categories of this paper are: q-alg
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The categories of this paper are: hep-lat
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The categories of this paper are: hep-th
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The categories of this paper are: stat
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The categories of this paper are: plasm-ph
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The categories of this paper are: q-bio
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The categories of this paper are: hep-ex
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The categories of this paper are: chao-dyn
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The categories of this paper are: supr-con
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The categories of this paper are: funct-an