Zero-divisor graphs of partially ordered sets

Let ( P , ≤ ) be a partially ordered set (poset, briefly) with a least element 0 and S ⊆ P . An element x ∈ P is a lower bound of S if s ≥ x for all s ∈ S . A simple graph G ( P ) is associated to each poset P with 0. The vertices of the graph are labeled by the elements of P , and two vertices x , y are connected by an edge in case 0 is the only lower bound of { x , y } in P . We show that if the chromatic number χ ( G ( P ) ) and the clique number ω ( G ( P ) ) are finite, then χ ( G ( P ) ) = ω ( G ( P ) ) = n + 1 in which n is the number of minimal prime ideals of P .