The Zero-divisor Graphs of Posets and an Application to Semigroups

Abstract

In this paper, we introduce the notion of a compact graph. We show that a simple graph is a compact graph if and only if G is the zero-divisor graph of a poset, and give a new proof of the main result in Halas and Jukl (Discrete Math 309:4584–4589, 2009) stating that if G is the zero-divisor graph of a poset, then the chromatic number and the clique number of G coincide under a mild assumption. We observe that the zero-divisor graphs of reduced commutative semigroups (rings) are compact, thus provide a large class of graphs G that could be realized as zero-divisor graphs of posets. In addition, using these results, we give some equivalent descriptions for the zero-divisor graphs of posets and reduced commutative semigroups with 0 respectively.