, Volume 29, Issue 3, pp 499506
First online:
Zero Divisor Graph of a Poset with Respect to an Ideal
 Vinayak JoshiAffiliated withDepartment of Mathematics, University of Pune Email author
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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.
Keywords
Zero divisor graph Clique number Chromatic number Annihilator Semiideal Ideal Prime ideal Semiprime idealMathematics Subject Classifications (2010)
Primary 05C15; Secondary 06A12 Title
 Zero Divisor Graph of a Poset with Respect to an Ideal
 Journal

Order
Volume 29, Issue 3 , pp 499506
 Cover Date
 201211
 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
 Authors

 Vinayak Joshi ^{(1)}
 Author Affiliations

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