, Volume 29, Issue 3, pp 499-506
Date: 10 May 2011

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.