Coloring of zero-divisor graphs of posets and applications to graphs associated with algebraic structures

TL;DR

This paper characterizes certain properties of zero-divisor graphs of finite posets, proves conjectures for these graphs and their complements, and applies these results to various algebraic structures, linking graph theory and algebra.

Contribution

It introduces new characterizations of zero-divisor graphs of finite posets and applies these to analyze graphs associated with algebraic structures, confirming conjectures in this context.

Findings

Zero-divisor graphs of finite posets are characterized as chordal and perfect.

Zero-divisor graphs and their complements satisfy the Total Coloring Conjecture.

Applications to graphs of rings, ideals, and subgroups demonstrate the utility of the poset approach.

Abstract

In this paper, we characterize chordal and perfect zero-divisor graphs of finite posets. Also, it is proved that the zero-divisor graphs of finite posets and the complement of zero-divisor graphs of finite $0$-distributive posets satisfy the Total Coloring Conjecture. These results are applied to the zero-divisor graphs of finite reduced rings, the comaximal ideal graph of rings, the annihilating ideal graphs, the intersection graphs of ideals of rings, and the intersection graphs of subgroups of cyclic groups. In fact, it is proved that these graphs associated with a commutative ring $R$ with identity can be effectively studied via the zero-divisor graph of a specially constructed poset from $R$.