The coloring of the regular graph of ideals

TL;DR

This paper investigates the properties of the regular graph of ideals of commutative rings, providing formulas for chromatic and clique numbers, especially for Artinian and reduced rings with multiple minimal prime ideals.

Contribution

It establishes the edge chromatic number and clique number formulas for the regular graph of ideals in specific classes of rings, advancing understanding of their graph-theoretic properties.

Findings

Edge chromatic number equals maximum degree for Artinian rings.

Formula for the clique number of the graph is provided.

Chromatic number and clique number are both n-1 for reduced rings with n minimal primes.

Abstract

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.