Zero Divisor Graph of a Poset with Respect to an Ideal
 Vinayak Joshi
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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 3cycles and 4cycles. 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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 Title
 Zero Divisor Graph of a Poset with Respect to an Ideal
 Journal

Order
Volume 29, Issue 3 , pp 499506
 Cover Date
 20121101
 DOI
 10.1007/s1108301192162
 Print ISSN
 01678094
 Online ISSN
 15729273
 Publisher
 Springer Netherlands
 Additional Links
 Topics
 Keywords

 Zero divisor graph
 Clique number
 Chromatic number
 Annihilator
 Semiideal
 Ideal
 Prime ideal
 Semiprime ideal
 Primary 05C15; Secondary 06A12
 Industry Sectors
 Authors

 Vinayak Joshi ^{(1)}
 Author Affiliations

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