Zero Divisor Graph of a Poset with Respect to an Ideal
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In this paper, we introduce the zero divisor graph GI(P) of a poset P (with 0) with respect to an ideal I. It is shown that GI(P) is connected with its diameter ≤3, and if GI(P) contains a cycle, then the core K of GI(P) is a union of 3-cycles and 4-cycles. Further, the chromatic number and clique number of GI(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.
KeywordsZero divisor graph Clique number Chromatic number Annihilator Semi-ideal Ideal Prime ideal Semiprime ideal
Mathematics Subject Classifications (2010)Primary 05C15; Secondary 06A12
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