Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: The coloring of the regular graph of ideals   The regular graph of ideals of the commutative ring $R$, denoted by
$\Gamma_{reg}(R)$, is a graph whose vertex set is the set of all non-trivial
ideals of $R$ and two distinct vertices $I$ and $J$ are adjacent if and only if
either $I$ contains a $J$-regular element or $J$ contains an $I$-regular
element. In this paper, it is shown that for every Artinian ring $R$, the edge
chromatic number of $\Gamma_{reg}(R)$ equals its maximum degree. Then a formula
for the clique number of $\Gamma_{reg}(R)$ is given. Also, it is proved that
for every reduced ring $R$ with $n(\geq3)$ minimal prime ideals, the edge
chromatic number of $\Gamma_{reg}(R)$ is $2^{n-1}-2$. Moreover, we show that
both of the clique number and vertex chromatic number of $\Gamma_{reg}(R)$ are
$n-1$, for every reduced ring $R$ with $n$ minimal prime ideals.

31
The categories of this paper are: ao-sci
0
The categories of this paper are: hep-ph
5
The categories of this paper are: gr-qc
6
The categories of this paper are: astro-ph
29
The categories of this paper are: funct-an
1
The categories of this paper are: cs
16
The categories of this paper are: stat
18
The categories of this paper are: econ
37
The categories of this paper are: supr-con
19
The categories of this paper are: eess
7
The categories of this paper are: hep-th
30
The categories of this paper are: q-alg
27
The categories of this paper are: comp-gas
11
The categories of this paper are: hep-lat
10
The categories of this paper are: q-bio
24
The categories of this paper are: dg-ga