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

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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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Authors and Affiliations

  1. Department of Mathematics, Soochow University, Suzhou, 215006, People’s Republic of China

    Dancheng Lu

  2. Department of Mathematics, Shanghai Jiaotong University, Shanghai, 200240, People’s Republic of China

    Tongsuo Wu

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  1. Dancheng Lu
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  2. Tongsuo Wu
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Correspondence to Dancheng Lu.

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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

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