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 Halaš 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.
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Lu, D., Wu, T. The Zero-divisor Graphs of Posets and an Application to Semigroups. Graphs and Combinatorics 26, 793–804 (2010). https://doi.org/10.1007/s00373-010-0955-4
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DOI: https://doi.org/10.1007/s00373-010-0955-4