Zero Divisor Graph of a Poset with Respect to an Ideal

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.