In this paper, we consider the crossing number and the chromatic number of the zero divisor graph Γ( ) 2 2 Z p to show that this type of zero divisor graphs is bipartite graph, and the smallest cycle in Γ( ) 2 2 Z p is of length four this implies that the girth is equal four. Index Terms-Bipartite graph, crossing number, girth, planar graph, zero divisor graph of the ring ( Z p 2 2 ).