Coloring of the annihilator graph of a commutative ring

Abstract

Let [Formula: see text] be a commutative ring with identity, and let [Formula: see text] be the set of zero-divisors of [Formula: see text]. The annihilator graph of [Formula: see text] is defined as the graph AG[Formula: see text] with the vertex set [Formula: see text], and two distinct vertices [Formula: see text] and [Formula: see text] are adjacent if and only if ann[Formula: see text]. In this paper, we study annihilator graphs of rings with equal clique number and chromatic number. For some classes of rings, we give an explicit formula for the clique number of annihilator graphs. Among other results, bipartite annihilator graphs of rings are characterized. Furthermore, some results on annihilator graphs with finite clique number are given.

Coloring in essential annihilating-ideal graphs of commutative rings

“…Although graph coloring is an old topic, it is still one of the most active areas in graph theory; for the most recent study in graph coloring see for instance [7], [10], [16], [19] and [20]. In addition to wide range of applications, the complexity of computations has caused considerable interest in characterizing these invariants for graphs associated with algebraic structures, some examples in this direction may be found in [3], [8], [12] and [14]. This paper is in this field and aims to investigate the the vertex and edge coloring in essential annihilating-ideal graphs of commutative rings.…”

On Perfect Co-Annihilating-Ideal Graph of a Commutative Artinian Ring

“…Therefore, the study of graphs associated with rings has attracted many researches. There are a lot of papers which apply combinatorial methods to obtain algebraic results in ring theory; for instance see [2,3,5,6,10,11] and [12].…”