Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: On coloring of fractional powers of graphs   For $m, n\in \N$, the fractional power $\Gmn$ of a graph $G$ is the $m$th
power of the $n$-subdivision of $G$, where the $n$-subdivision is obtained by
replacing each edge in $G$ with a path of length $n$. It was conjectured by
Iradmusa that if $G$ is a connected graph with $\Delta(G)\ge 3$ and $1<m<n$,
then $\chi(\Gmn)=\omega(\Gmn)$. Here we show that the conjecture does not hold
in full generality by presenting a graph $H$ for which
$\chi(H^{3/5})>\omega(H^{3/5})$. However, we prove that the conjecture is true
if $m$ is even. We also study the case when $m$ is odd, obtaining a general
upper bound $\chi(\Gmn)\leq \omega(\Gmn)+2$ for graphs with $\Delta(G)\geq 4$.

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The categories of this paper are: hep-ph
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The categories of this paper are: math-ph
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The categories of this paper are: solv-int
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The categories of this paper are: bayes-an
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The categories of this paper are: hep-lat
22
The categories of this paper are: chao-dyn
30
The categories of this paper are: q-alg
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The categories of this paper are: supr-con
35
The categories of this paper are: mtrl-th
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The categories of this paper are: eess
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The categories of this paper are: alg-geom
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The categories of this paper are: q-fin
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The categories of this paper are: chem-ph
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The categories of this paper are: cs
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The categories of this paper are: funct-an
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The categories of this paper are: hep-th