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Zero Divisor Graph of a Poset with Respect to an Ideal

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Abstract

In this paper, we introduce the zero divisor graph G I (P) of a poset P (with 0) with respect to an ideal I. It is shown that G I (P) is connected with its diameter ≤3, and if G I (P) contains a cycle, then the core K of G I (P) is a union of 3-cycles and 4-cycles. Further, the chromatic number and clique number of G I (P) are shown to be equal. This proves a form of Beck’s conjecture for posets with 0. The complete bipartite zero divisor graphs are characterized.

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Authors and Affiliations

  1. Department of Mathematics, University of Pune, Pune, 411007, India

    Vinayak Joshi

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  1. Vinayak Joshi
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Correspondence to Vinayak Joshi.

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Dedicated to Professor N. K. Thakare on his 71st birthday.

This research is supported by Board of College and University Development, University of Pune, via the project SC-66.

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Joshi, V. Zero Divisor Graph of a Poset with Respect to an Ideal. Order 29, 499–506 (2012). https://doi.org/10.1007/s11083-011-9216-2

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  • DOI: https://doi.org/10.1007/s11083-011-9216-2

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